Topological derivatives for shell structure optimization considering crash loadcases with material nonlinearities and large deformations

Abstract

The topological derivatives for shell structures in dynamic loadcases, which are often used in lightweight designs, are developed. These sensitivities consider the highly nonlinear material behavior and large deformations of the structure. The aim of the investigation is the support of an efficient topology optimization for crashloaded structures. The considered functionals can in general be described by displacements, velocities and accelerations. In particular, the semi-analytical sensitivity calculations for the internal deformation energy and for single point displacements in arbitrary directions in the shell structure are presented. These functionals cover the most important functions for the development of crash structures. Using the material derivative in combination with the adjoint method, a terminal value problem has to be solved. Depending on whether first differentiation and then discretization in the time domain is performed or first discretization and then differentiation, a separate solution scheme for the adjoint is derived. Both schemes are presented in a comparable description. For the numerical evaluation of the topological derivatives, the basic equations for large deformed shell elements are included as well as a phenomenological material interpolation, which represents the tangential material behavior resulting from plastic strain and isotropic hardening. All assumptions made for the derivation are described in detail and their influences are discussed. The elaborated equations and the procedure for the calculation of the topological derivatives are applied, discussed and checked for plausibility on two clearly defined loading conditions for the internal deformation energy and single point displacements on different locations.

1 Introduction

Crashworthy body development serves to improve the passive safety of automotives, trains and aircrafts. The absorbing crash structures are mostly shell-like structures. For this reason, the consideration of the mechanics of shells is elementary important. The finite element method is used to simulate the crash behavior. With the simulation, the targeted force paths within the body can be ensured and deformations in areas requiring protection can be avoided. Crashworthiness optimization methods are based on these crash simulations.

In the early phase of the development process of structures, topology optimization is interesting, where the location and arrangement of structural elements are determined so that one can design a basic idea of the crashworthy structure in the first place. The fact, that such a crash is a highly nonlinear process complicates the determination of sensitivities for the optimization. We generally categorize nonlinearities into three groups:

• Geometric nonlinearities are large displacements and strains as well as bifurcation and snap-through problems.

• Nonlinear material behavior is, for example, that permanent deformations occur due to plastic flow.

• If the boundary conditions change during deformation, such as through contact with surrounding components, this is referred to as nonlinear boundary conditions.

Besides these nonlinearities, the crash is also a transient process where all properties change dynamically. In summary, such a crash is the most complicated continuum mechanics imaginable for which we would also like to have sensitivities for optimization.

The first topology optimization with sensitivities considering structures under crash-load is published in Pedersen (2003) and Pedersen (2004). The design space is filled with beam elements and the optimization objective is to achieve a given force-displacement curve. The heights of the beam elements serve as design variables for the optimization. The sensitivities are computed, but without taking inertia effects into account.

In Michaleris et al. (1994) the general sensitivity calculation with the density method for nonlinear transient systems with both the direct and adjoint methods was developed. Material nonlinearities and large displacements and strains are considered in the transient calculation, but no inertia effects. For the first time, the final value problem for the adjoint calculation arises, to which any transient sensitivity calculation with the adjoint method leads. Using an elasto-plastic material law, the derivative of internal forces with respect to the displacement—the tangent stiffness—is also required here for solving the adjoint state equation for the first time. This development is closely related to nonlinear material behavior and is the basis of any recent research related to the adjoint method. Based on this sensitivity calculation, Wallin et al. (2016) used geometric nonlinearities and elasto-plastic material behavior to maximize plastic work with the density method.

Nonlinear stiffness optimizations were first performed by Wallin et al. (2018) with the adjoint method. As a result, the tangent stiffness is also used here. However, the loadcase calculation was not performed using transient methods, which means that inertia effects could not be taken into account. Based on this work and using the aforementioned preliminary work of Michaleris et al. (1994) and Wallin et al. (2016), a concept for maximizing plastic work with large displacements and transient loads while taking inertia into account was finally developed by Ivarsson et al. (2018). The material formulation in the form of a viscoplastic material law for large displacements and strains is coupled to the sensitivity calculation. Although, the loading of a reduced front end by a frontal pole impact had to be modeled here with prescribed displacements, since no contact formulations were used. Nevertheless, different geometries could be generated with different prescribed velocities.

Some topology optimization methods exist that rely on heuristic surrogate sensitivities. That is, sensitivities for the typical objectives in the crash can’t yet be calculated and one has to fall back to either equivalent static loadcases (such as the Equivalent Static Load Method ESLM, cf. Choi and Park (2002) and Park (2011)) or the internal energy density (such as the Hybrid Cellular Automaton method HCA (Patel et al. 2009), the Hybrid Cellular Automata for Thin Walled Structures HCATWS (Hunkeler 2013; Zeng and Duddeck 2017) and the work of Forsberg and Nilsson (2007)). With the first approach of the Graph and Heuristic based Topology optimization (GHT) from Olschinka and Schumacher (2008), Ortmann and Schumacher (2013) and Ortmann et al. (2021), laterally loaded extruded profiles were optimized in the cross section. For example, the vehicle rocker in the lateral impact is iteratively filled with new walls. The locations of the new walls are determined using heuristics derived from expert knowledge. Mathematical optimization is then performed with numerical sensitivity determination to improve thicknesses and wall positions.

On one hand, the use of analytical sensitivities has a great potential to improve the structural functionality with topology optimization with acceptable numerical effort. On the other hand, reliable crash simulations have strong modeling requirements, that cannot be satisfied with intermediate density modeling. Desirable for the topology optimization of crash loaded structures is a modeling and sensitivity determination that does not require intermediate densities and at the same time allows a calculation with a clear geometry. The topological derivative can provide a solution here.

The idea of the topological derivative is to gain the sensitivity of a functional $\mathcal {J}$ with respect to a point X₀ in the body ${\varOmega }$ by introducing a cut-out $c_r(\mathbf{{X}}_0)$ with the dimensioning parameter $r$ and thus by directly changing the topology class as shown in Fig. 1.

The differential quotient for the functional $\mathcal {J}$ is called topological derivative of the functional at the point $\mathbf{{X}}_0$. This differential quotient is formed by the difference of the functional on the remaining area and the functional on the area without cut-out and divided by the volume of the cut-out

$$\begin{aligned} \mathcal {T}\!\mathcal {J}(\mathbf{{X}_0)} := \lim _{r\downarrow 0} \frac{\mathcal {J}\left( {\varOmega }\setminus \overline{c_r(\mathbf{{X}_0)} }\right) - \mathcal {J}\left( {\varOmega }\right) }{\left| \overline{c_r(\mathbf{{X}_0})}\right| }\,. \end{aligned}$$

(1)

Fig. 1

Body with cut-out

With the transition to the limit, ultimately only one point of the body is left out. The topological derivative is therefore understood as a sensitivity for the design variable “Point $\mathbf{{X}}_0$ not there”. If the sensitivity is less than zero ($\mathcal {T}\!\mathcal {J}\left( \mathbf{{X}}_0\right) <0$), an infinitesimal cut-out would cause a reduction of the functional. If the sensitivity is greater than zero ($\mathcal {T}\!\mathcal {J}\left( \mathbf{{X}}_0\right) >0$), the cut-out would cause an increase of the functional. An evaluation of this sensitivity for all points of the design space provides the information about the locations where the insertion of a cut-out is useful for the optimization problem. The calculation formula for the topological derivative can be developed with the so-called material derivative (cf. Sokołowski and Zolesio (1992)) in combination with the adjoint method. In the special case of static linear elasticity with the help of Airy’s stress function and the complex potentials approach, a closed formula for the calculation in terms of stresses and strains and adjoint stresses and strains can be established.

The development of the topological derivation goes back to the bubble method of Eschenauer et al. (1994) and is described there as a positioning criterion. The term “topological derivative” comes from Sokołowski and Zochowski (1999). Initially, the topological derivative also classically considered the compliance. In Schumacher (1995), a failure functional based on the von Mises comparative stress is developed, this is taken up again in a more general form in Amstutz and Novotny (2010). A global displacement functional can be found in Sokołowski and Zochowski (1999). The topological derivative for compliance and grip** mechanisms can be found in Amstutz (2011) and Lopes and Novotny (2016). Especially for linear elasticity, Lewinski and Sokołowski (2006) developed the Topological Derivative for spherical shells. Also contact between components and its influence on the topological derivative and on sensitivities in general, respectively, has already been studied by Giusti et al. (2015) und Lopes et al. (2017) for linear elasticity and in Lawry and Maute (2018) and Fernandez et al. (2020) for nonlinear elasticity. In Fernandez et al. (2020) even sensitivities for the contact force are derived.

The mathematical theoretical basis of the topological derivative is presented in Novotny and Sokołowski (2013) in-depth. A compilation of application areas can be found in the book Applications of the Topological Derivative Method by Novotny et al. (2019a) and the corresponding articles (Novotny et al. 2019b, c, d). Only exemplary application areas such as electromechanics (e. g. Gangl et al. (2015)) and fluid mechanics (Duan and Li (2015)) are mentioned here.

In this work, the nonlinear transient topological derivative is developed. Based on this, a numerical calculation scheme is developed specifically for shell structures.

This publication aims at pushing the constraints imposed on the Topological Derivative by elastic material behavior further towards nonlinear optimization. This is understood for nonlinear material behavior, large deformations as well as for transient loads. In particular, due to the displacement states that occur after buckling phenomena, there is no uniqueness of the underlying initial boundary value problem and thus no uniqueness of the sensitivity. No mathematical theorems are postulated in this paper and, accordingly, no proofs are given. Rather, this paper is intended as an invitation to mathematicians to approach this subject area and to engineers such as finite element software developers to implement the algorithms.

In Sect. 2, the basics of nonlinear structure simulation are summarized. The solution of the discrete initial boundary value problem is done with implicit time integration. The definition of two functionals, used for crashworthy body development is given in Sect. 3. The general definition of the topological derivative and its description through the material derivative is presented in Sect. 4. The computation of the material derivative is done according to two equivalent schemes with the help of the adjoint method in Sect. 5 and completes the analytical part of this work. The specific application for shell models for the design of sheet metal components is discussed in Sect. 6. As an example, for an elasto-plastic material model, a material interpolation is developed and the semi-analytical topological derivative is calculated. The application and testing of the plausibility of the developed computational models of the nonlinear transient topological derivative is carried out on a practical example in Sect. 7.

2 Nonlinear continuum mechanics

First, we account for underlying kinematics and the nonlinear transient initial boundary value problem, that describes the nonlinear continuum mechanics in the mathematical way. Then we give a brief summary of the numerical solution that is considered in this paper.

2.1 Weak form of the mechanical initial boundary value problem

For the general continuum mechanics, we need to cover large displacements and material nonlinearities and therefore distinguish the current configuration of a body ${\varOmega }_t$, where the points of the body are placed in space for the time $t$, from the initial configuration ${\varOmega }$, where the body is undeformed. A material point in the initial configuration is described by the upper-case vector X. With the deformation and motion of the body, this material point undergoes a displacement u and is then located at a point in the current configuration that is described by the lower-case vector x:

$$\begin{aligned} \mathbf{{u}}\left(\mathbf{{X}}, t\right) = \mathbf{{x}}-\mathbf{{X}}\, . \end{aligned}$$

The deformation gradient $\varvec{F}$ relates the tangent vectors of the initial and the current configuration to each other (Wriggers 2008)

$$\begin{aligned} \varvec{F}:= {{\,\textrm{Grad}\,}}\mathbf{{x}}= {{\,\textrm{Grad}\,}}{\mathbf{u}} + \varvec{1}\, . \end{aligned}$$

(2)

The Green-Lagrange strain tensor $\varvec{E}$ is related to the displacement gradient by

$$\begin{aligned} \varvec{E}:= \frac{1}{2}({{\,\textrm{Grad}\,}}\mathbf{{u}} + {{\,\textrm{Grad}\,}}\mathbf{{u}}^T + {{\,\textrm{Grad}\,}}\mathbf{{u}}^T {{\,\textrm{Grad}\,}}\mathbf{{u}}) \, . \end{aligned}$$

Under the assumption of small displacements, the quadratic term is neglected and the infinitesimal strain tensor is defined as

$$\begin{aligned} \varvec{\varepsilon }:= \frac{1}{2}({{\,\textrm{Grad}\,}}\mathbf{{u}} + {{\,\textrm{Grad}\,}}\mathbf{{u}}^T )\, . \end{aligned}$$

The second Piola-Kirchhoff stress tensor $\varvec{S}$ results from a complete transformation of the Cauchy stress $\varvec{\sigma }$ to the initial configuration. As a wide range of material modeling exist in (commercial) finite element solvers, we do not focus on a particular constitutive model so far. For the calculation in Sect. 6 a phenomenological description of an elasto-plastic material behavior with a piecewise linear strain hardening curve is used in a phenomenological description in terms of the Cauchy stress tensor and the effective plastic strain.

The boundary of the body, that is loaded with a stress vector $\varvec{\uptau }$, is called Neumann-boundary and denoted with ${\varGamma }_N$. On the Dirichlet-boundary ${\varGamma }_D$, the displacement is prescribed with $\mathbf{{u}} = \bar{\mathbf{{u}}}$. With $c_d$ we denote a velocity depending specific linear dam** which is mainly introduced in the discretized system to represent loss by traction and viscosity in the material in the numerical solution. We introduce this here to compare the adjoint solution schemes at the end of Sect. 5. The density of the body is denoted with $\varrho$.

The solution of the initial boundary value problem is a vector-valued displacement function $\mathbf{{u}}\left( \mathbf{{X}}, t\right) : {\varOmega }\times \left[ 0, T\right] \rightarrow \mathbb {R}^d$, that satisfies the Dirichlet boundary conditions. The time $T$ is the final time, when the mechanical systems exceeds validity. The first derivative of the solution with respect to time is the velocity $\frac{\partial }{\partial t}\varvec{\textrm{u}} = \dot{\varvec{\textrm{u}}}\left( \varvec{\textrm{X}}, t\right)$, the second derivative is the acceleration $\frac{\partial ^2}{\partial t^2}\mathbf{{u}} = \ddot{\mathbf{{u}}}\left(\mathbf{{X}}, t\right)$.

The following notation of the dual pairing is used to clarify that the functional $G$ is mapped with the test function $\varvec{\phi }$ to the space of $\mathbb {R}$-valued functions.¹

The initial boundary value problem of the nonlinear continuum mechanics with the initial condition $\dot{\mathbf{{u}}}\left( \, \cdot \,, t= 0\right) = \mathbf{\bar{{\dot{{u}}}}}$ in the weak form in the initial configuration reads as

$$\begin{aligned} &\bigg \langle G\big (\mathbf{{u}}\left( {\varOmega }\right) , \dot{\mathbf{{u}}}\left( {\varOmega }\right) , \ddot{\mathbf{{u}}}\left( {\varOmega }\right) , {\varOmega }\big ) \,, \, \varvec{\phi } \bigg \rangle := \nonumber \\ &\underbrace{\int \limits _{{\varOmega }}\varvec{F}\varvec{S} \cdot {{\,\textrm{Grad}\,}}\varvec{\phi } \,\text {d}{\varOmega }}_{\approx \varvec{\phi }^T \mathbf{{r}} \left( \mathbf{{u}}\right) } + \underbrace{\int \limits _{{\varOmega }}c_d \dot{\mathbf{{u}}} \cdot \varvec{\phi } \,\text {d}{\varOmega }}_{\approx \varvec{\phi }^T \mathbf{{C}}\dot{\mathbf{{u}}}} + \underbrace{\int \limits _{{\varOmega }}\varrho \ddot{\mathbf{{u}}} \cdot \varvec{\phi } \,\text {d}{\varOmega }}_{\approx \varvec{\phi }^T \mathbf{{M}}\ddot{\mathbf{{u}}}}\nonumber \\&\quad \underbrace{- \int \limits _{{\varOmega }}\mathbf{{f}} \left( t\right) \cdot \varvec{\phi } \,\text {d}{\varOmega }- \int \limits _{{\varGamma }_N} \varvec{\uptau } (t) \cdot \varvec{\phi } \,\text {d}{\varGamma }}_{\approx -\varvec{\phi }^T \mathbf{{q}}} = 0 \,. \end{aligned}$$

(3)

The first term denotes the internal virtual work, the second the dam**, the third term the inertia and the last two terms the virtual work of the applied loads with the specific volume force $\mathbf{{f}}$. The test function $\varvec{\phi } (\mathbf{{X}}, t):{\varOmega }\times \left[ 0, T\right] \rightarrow \mathbb {R}^d$ vanishes on the Dirichlet boundary ${\varGamma }_D(t)$: $\varvec{\phi } \in \left\{ \varvec{\phi }\,|\,\varvec{\phi } = \mathbf{{0}} \text { on } {\varGamma }_D(t)\right\}$.

2.2 Discretization in space and time

In general, it is not possible to solve the initial boundary value problem analytically. With a finite element discretization (the abbreviations are given under the brackets below the integral terms in (3)) and a time integration scheme, one gets the solution discretized in time and space. In this paper we refer to the implicit time integration, as it makes the calculation of the adjoint much easier by the existence of the tangent stiffness matrix. In principle, the solution may also be gained by explicit time integration.

As we use the solution scheme in particular for the derivation of the adjoint, we give here the most important formulas for the implicit time integration, which is solved with Newton’s method. For the displacement and the velocity at the time ${t^{(k+1)}}$, the so-called Newmark approximation

$$\begin{aligned} {\mathbf{{u}}^{(k+1)}}&= {\mathbf{{u}}^{(k)}} + \left( \varDelta {t^{(k+1)}}\right) {\dot{\mathbf{{u}}}^{(k)}} \nonumber \\&+ \frac{\left( \varDelta {t^{(k+1)}}\right) ^2}{2}\left( (1-2 \beta ){\ddot{\mathbf{{u}}}^{(k)}} + 2 \beta {\ddot{\mathbf{{u}}}^{(k+1)}}\right) \, ,\end{aligned}$$

(4)

$$\begin{aligned} {\dot{\mathbf{{u}}}^{(k+1)}}&= {\dot{\mathbf{{u}}}^{(k)}}\nonumber \\&+ \left( \varDelta {t^{(k+1)}}\right) \left( (1-\gamma ){\ddot{\mathbf{{u}}}^{(k)}} + \gamma {\ddot{\mathbf{{u}}}^{(k+1)}}\right) \end{aligned}$$

(5)

is inserted into the discretized balance of the linear momentum. The free constants $0 \le \beta \le 0.5$ and $0\le \gamma \le 1$ are chosen as $\gamma =0.5$ and $\beta =0.25$ in this work. The timestep size $\varDelta {t^{(k+1)}} = {t^{(k+1)}} - {t^{(k)}}$ can vary from time step to time step.

It is assumed, that for all points in time ${t^{(i)}}$ from $i=0$ up to k the displacement, velocity and acceleration are determined and the next balance for the time ${t^{(k+1)}}$ is sought up to the time ${t^{(m)}}=T$. The resulting nonlinear equation – nonlinear in the vector of the internal forces $\mathbf{{r}}\left( {\mathbf{{u}}^{(k+1)}}\right)$ —is solved with Newton’s method. The differentiation of the internal forces with respect to the displacement is denoted with the so-called tangential stiffness matrix

$$ {\mathbf{K}}_{{\mathbf{T}}} \left( {^{{(j)}} {\mathbf{u}}^{{(k + 1)}} } \right) = \frac{{\partial {\mathbf{r}}}}{{\partial {\mathbf{u}}^{{(k + 1)}} }}\Bigg|_{{^{{(j)}} {\mathbf{u}}^{{(k + 1)}} }} , $$

where the left upper index describes the iteration steps of Newton’s algorithm.

3 Definition of functionals

For the derivation of the topological derivative, we first consider a general functional that can depend on the displacement as well as on the velocity and acceleration. Likewise, an integration of an inner function $z\left( \mathbf{{u}}, \dot{\mathbf{{u}}}, \ddot{\mathbf{{u}}} \right)$ over the entire body is provided and a time integral

$$\begin{aligned} \mathcal {J}\left( \mathbf{{u}}({\varOmega }), \dot{\mathbf{{u}}}({\varOmega }), \ddot{\mathbf{{u}}}({\varOmega }), {\varOmega }\right) = \int \limits _0^T\int \limits _{{\varOmega }}z\left( \mathbf{{u}}, \dot{\mathbf{{u}}}, \ddot{\mathbf{{u}}} \right) \,\text {d}{\varOmega }\,\text {d}t\, . \end{aligned}$$

(6)

These conditions are however not necessary for the computability.

In principle, all functionals that are also used in linear statics can be extended to dynamic loads. In terms of crashworthy body design, the sensitivity of the internal energy should be determined so that the structure can be selectively softened to convert the external force and kinetic energy into plastic strain energy. In combination with the displacement of a single point, it should then be possible to develop classic crash requirements such as minimum acceleration of the vehicle occupant during impact while ensuring survival space.

The internal energy is continuously defined as

$$\begin{aligned} \mathcal {J}_{\sigma }:= \int \limits _{{\varOmega }}\int \limits _0^T\varvec{F}\varvec{S} \cdot {{\,\text{Grad}\,}}\dot{\mathbf{{u}}} \,\text {d}t\,\text {d}{\varOmega }\, . \end{aligned}$$

(7)

The discretization in space is given through the scalar product of the vector of the internal forces with the velocity. The time weighting factors ${w^{(k)}}$ may be chosen free, we make use of the trapezoidal integration weights to stay consistent with the Newmark-constants $\beta$ and $\gamma$. The discretized functional is denoted with the lower-case j to distinguish between continuous and discretized formulation

$$\begin{aligned} j_{\sigma }= {\sum _{k=0}^m{w^{(k)}}} {{\dot{\textbf{u}}^{(k)^{T}}}} \mathbf{{r}}\left( {\mathbf{{u}}^{(k)}}\right) \, . \end{aligned}$$

(8)

To detect the maximum displacement of the point $\mathbf{{X}}_*$ in direction $\mathbf{{v}}$ over time, the single point displacement functional is constructed p-norm-like

$$\begin{aligned} \mathcal {J}_u= \left( \frac{1}{T} \int \limits _0^T\left( \big . \mathbf{{u}}(\mathbf{{X}}, t) \cdot \mathbf{{v}} \big |_{\mathbf{{X}}= \mathbf{{X}_*}} \right) ^{p} \,\text {d}t\right) ^{\frac{1}{p}} \, . \end{aligned}$$

(9)

In the spatial discretization, the direction vector $\mathbf{{v}}$ is built into the vector $\mathbf{{l}} = \left( 0 \cdots \mathbf{{v}}^T \cdots 0 \right) ^T$ of the degrees of freedom, that form the displacement direction

$$\begin{aligned} j_u= \left( \frac{1}{T}\sum _{k = 0}^{m}{w^{(k)}}\left( \mathbf{{l}}^T{{\mathbf{u}}^{(k)}}\right) ^p\right) ^{\frac{1}{p}} \, . \end{aligned}$$

(10)

A classical use case for the design of a crashloaded structure is to convert as much as possible of the kinetic energy within the structure into plastic strain while not exceeding a given force level. With the functionals presented in the paper, a modeling of the task would be possible as follows: Maximize the internal energy in the structure. If the deformation path is restricted by usage of the single point displacement, the maximum force level is indirectly given for energy conservation reasons.

4 Definition of the topological derivative

The topological derivative as defined in (1) is an indicator for “point of the set is valuable” or just less valuable to achieve an optimization goal or to still be able to comply with a restriction. Thus, this is a sensitivity to the design variable “no material at point $\mathbf{{X}}_0$”. This cutting out is a non-smooth operation in the sense of changing the shape of the body, since the topology class changes.

We denote with $c_r(\mathbf{{X}}_0)$ a general cut-out function. The closure of $c_r(\mathbf{{X}}_0)$ is denoted with $\overline{c_r(\mathbf{{X}}_0)}$. The volume $\left| \overline{c_r(\mathbf{{X}}_0)}\right|$ vanishes for the limit passage $r\rightarrow 0$. The boundary of the cut-out is denoted by ${\varGamma }_{r}$. The general definition of the cut-out function allows arbitrary cut-out shapes, for example an orientation in space with elliptical cut-outs. Thus, also an artificial orthotropy can be introduced into the system. In this work, mainly two cut-out shapes are considered. For full-dimensional planar ($d=2$) and spatial ($d=3$) modeling as in Figs. 1 and 2a, spherical cut-outs $B_r(\mathbf{{X}}_0)$ with radius $r>0$ around $\mathbf{{X}}_0 \in {\varOmega }$ are defined by

$$\begin{aligned} B_r(\mathbf{{X}}_0) := \left\{ \mathbf{{Y}} \in {\varOmega }\, | \, ||\mathbf{{Y}}-\mathbf{{X}}_0||_2 < r\right\} \, . \end{aligned}$$

Cylindrical holes $C_{rh}(\mathbf{{X}}_0)$ (cf. Fig. 2b) are used in shell models and therefore of special interest in Sect. 6. The cylindrical cut-out also depends on the shell thickness $h$. We assume that for shell models the perturbation point is placed in the midsurface $\mathbf{{X}}_0\in \mathcal {M}$:

$$\begin{aligned} C_{rh} (\mathbf{{X}}_0) := \left\{ \mathbf{{Y}} \in {\varOmega }\, | \, ||\mathbf{{Y}}_{\mathcal {M}}-\mathbf{{X}}_0||_2 < r\right\} \, . \end{aligned}$$

The set with the cut-out is abbreviated to ${\varOmega }_r:= {\varOmega }\setminus c_r(\mathbf{{X}}_0)$ for clear notation of the derivatives and integrals.

Fig. 2

Dimensions of the cut-out

The following derivation holds for both spherical and cylindrical cut-outs, so we will continue with the general cut-out function $c_r(\mathbf{{X}}_0)$. The representation in the initial configuration is intended to facilitate the evaluation of the formulas later, since round sections remain round even in the case of deformation. Therefore, the initial configuration is also retained in the following derivations.

The topological derivative of the functional $\mathcal {J}$ as defined in (6) with respect to a cut-out $c_r(\mathbf{{X}}_0)$ at the point $\mathbf{{X}}_0$ is defined by the differential quotient and can be calculated using the material derivative (Novotny and Sokołowski 2013), which can be found in the second line of the following equation:

$$\begin{aligned} \mathcal {T}\!\mathcal {J}(\mathbf{{X}_0}) &= \lim _{r\downarrow 0} \Bigg ( \frac{\,\text {d}\left( \left| \overline{c_r(\mathbf{{X}}_0)}\right| \right) }{\,\text {d}r}\Bigg )^{-1} \nonumber \\& \qquad \quad \; \cdot \Bigg (\underbrace{\int \limits _{{\varOmega }_r} \frac{\partial }{\partial r} z\,\text {d}{\varOmega }}_{\text {impl. }\left( \frac{\partial \mathcal {J}}{\partial r}\right) _I} \underbrace{- \int \limits _{{\varGamma }_r}z\,\text {d}{\varGamma }}_{ \text {expl. }\left( \frac{\partial \mathcal {J}}{\partial r}\right) _E} \Bigg ) \,. \end{aligned}$$

(11)

The fact that the functional is required here as an integral over the whole body is only due to the definition with the material derivative. If there is no integral over the whole body, as is the case with the displacement of the single point $\mathcal {J}_u$, the explicit derivative for this functional is omitted. For functionals, that are not defined by an integral over the body but through an integral on the boundary, the material derivative can be found in Sokołowski and Zolesio (1992).

The material derivative consists of two integrals: One integral on the area with cut-out—in the following called implicit derivative—and one integral on the cut-out boundary—in the following called explicit derivative.

The derivative of the cut-out volume with respect to the parameter $r$ yields the surface measure of the cut-out, i. e.

$$\begin{aligned} \frac{\,\text {d}}{\,\text {d}r}\left| \overline{B_r(\mathbf{{x}}_0)}\right| = 2(d-1) r^{d-1} \pi \quad \text {with} \quad d= 2, 3 \end{aligned}$$

and

$$\begin{aligned} \frac{\,\text {d}}{\,\text {d}r}\left| \overline{C_{rh}(\mathbf{{x}}_0)}\right| = 2 \pi rh \, . \end{aligned}$$

(12)

Especially the calculation of the implicit derivatives in (11) is generally not possible. With use of the so-called adjoint method, which will be shown in Sect. 5, the implicit integral may be eliminated. The evaluation of the remaining explicit derivative will be discussed in Sect. 6.

5 Adjoint method

The idea is to extend the functional with the equilibrium Eq. (3). Specifically, the equilibrium equation is added to the functional as an equality constraint with its Lagrangemultiplier ${\varvec{{\lambda}}}$ instead of the arbitrary test function $\varvec{\phi }$. Then, this extended functional is differentiated with the material derivative. The resulting terms which explicitly depend on the dimensioning parameter $r$, the boundary integrals on ${\varGamma }_r$, have to be evaluated. For the terms which implicitly depend on the dimensioning parameter, for example via the chain rule, the multiplier ${\varvec{{\lambda}}}$ is chosen in such a way that the implicit derivative disappears. This multiplier is then called an adjoint or adjoint solution. A general functional analysis approach to the adjoint method can be found in Hinze et al. (2009).

Two approaches for the adjoint solution are known from the literature: The sequence of the differentiation with respect to the design parameter and of the discretization in the time domain can be performed as “first differentiate then discretize” or as “first discretize then differentiate”. In the next two sections, both variants will be presented.

5.1 First differentiate then discretize

Coming from the pure mathematical definition one would only see the continuous description of the functional, of the weak form of the equation of motion and of the topological derivative. The intuitive solution scheme for the adjoint is then derived via “first differentiate then discretize”. That means, first, the material derivative of the augmented functional is built. The emerging adjoint equation of motion is subsequently solved with time discretization. The use of partial integration in the time domain to gain the adjoint equation of motion is a well-known trick in wave propagation optimization (see Dahl et al. (2008)). In Weider and Schumacher (2019) this trick was adapted for the nonlinear transient case of the topological derivative. In addition, in Weider (2021) also the dependency on the acceleration for the functional is considered.

We start with the a general functional and extend that with the weak form of the equation of motion (3). In particular, since the equilibrium is satisfied at all times, it is satisfied as an integral over time. Note, that the Lagrange multiplier ${\varvec{{\lambda}}}$ replaces the test function $\varvec{\phi }$. As we need first the material derivative to get the topological derivative and this material derivative is in fact a shape derivative with respect to the parameter $r$, we consider both, the functional as well as the equilibrium on an infinitesimal perturbed body ${\varOmega }_r$:

$$\begin{aligned} \hat{\mathcal {J}}:&= \mathcal {J}\left( \mathbf{{u}}({{\varOmega }_r}), \dot{\mathbf{{u}}}({{\varOmega }_r}), \ddot{\mathbf{{u}}}({{\varOmega }_r}), {{\varOmega }_r} \right) \nonumber \\ & \quad+ \int \limits _0^T\Big \langle G\left( \mathbf{{u}}({{\varOmega }_r}), \dot{\mathbf{{u}}}({{\varOmega }_r}), \ddot{\mathbf{{u}}}({{\varOmega }_r}), {{\varOmega }_r} \right) \,, \, {\varvec{\lambda}} \Big \rangle \,\text {d}t\,. \end{aligned}$$

(13)

We now consider in particular the material derivative of the extended functional $\hat{\mathcal {J}}$. The Neumann boundary ${\varGamma }_N$ with the applied stresses is independent of the dimensioning parameter $r$ of the cut-out. Therefore, the stress load no longer appears in the explicit derivation. The material derivative is split into the explicit and implicit derivatives:

$$\begin{aligned} \frac{\partial \hat{\mathcal {J}}}{\partial r}&=\left( \frac{\partial \mathcal {J}}{\partial r}\right) _E - \int \limits _0^T\int \limits _{{\varGamma }_r}\varvec{F}\varvec{S} \cdot {\varvec{\nabla}} {\varvec{\lambda}} + c_d\frac{\partial \mathbf{{u}}}{\partial t} \cdot {\varvec{\lambda}} \nonumber \\&\quad \underbrace{\qquad \qquad \qquad \qquad + \varrho \frac{\partial ^2 \mathbf{{u}}}{\partial t^2} \cdot {\varvec{\lambda}} - \mathbf{{f}}(t) \cdot {\varvec{\lambda}} \,\text {d}{\varGamma }\,\text {d}t}_{\text{ expl. } \text{ derivative }} \nonumber \\&\quad + \left( \frac{\partial \mathcal {J}}{\partial r}\right) _I + \bigg \langle \frac{\partial }{\partial \mathbf{{u}}} \int \limits _0^T\int \limits _{{\varOmega }_r}\varvec{F}\varvec{S} \cdot {\varvec{\nabla}} {\varvec{\lambda}} + c_d \dot{\mathbf{{u}}} \cdot {\varvec{\lambda}} \bigg . \nonumber \\&\quad \underbrace{\qquad \qquad \qquad \quad \bigg . + \varrho \ddot{\mathbf{{u}}} \cdot {\varvec{\lambda}} - \mathbf{{f}}(t) \cdot {\varvec{\lambda}} \,\text {d}{\varOmega }\,\text {d}t\, , \, \frac{\partial \varvec{\text{u}}}{\partial r}\bigg \rangle }_{\text{ impl. } \text{ derivative }} \, . \end{aligned}$$

(14)

Just like the general test function $\varvec{\phi }$, the adjoint has to vanish on the Dirichlet boundary ${\varGamma }_D$ of the body: ${\varvec{\lambda}} \in \left\{\varvec{{\lambda}}\,|\,{\varvec{\lambda}} = \mathbf{{0}} \text { on } {\varGamma }_D(t)\right\}$. For the adjoint ${\varvec{\lambda}}$ it is required, that the implicit derivative vanishes. Like the displacement, this adjoint is also a time-dependent, vector-valued function.

The infinitesimal cut-out $\lim _{r\downarrow 0} c_r(\mathbf{{X}}_0)$ is a null set with respect to the integration measure $\,\text {d}{\varOmega }$. It follows that any integral on the infinitesimal section is zero: $\lim _{r\downarrow 0} \int _{c_r(\mathbf{{X}}_0)} f \,\text {d}{\varOmega }= 0$. Since the integration is additive ($\int _{\varOmega }= \int _{{\varOmega }_r} + \int _{c_r(\mathbf{{X}}_0)}$), the implicit derivatives can be defined as an integral over the whole body without perturbation.

For a clearer visibility, we continue with spatial discretized forms of the body integrals for the implicit derivatives. The spatial discretization of the tangential stiffness ${\mathbf{{K}}}_{\mathbf{{T}}}^{(k)}$, the dam** matrix $\mathbf{{C}}$ and the mass matrix $\mathbf{{M}}$ are taken from the discretization and solution of the initial boundary value problem.

To determine the adjoint we need a trick. Adapted from Dahl et al. (2008), the dam** term, the inertia term and the implicit derivative of the functional are rewritten with partial integration in the time domain. Exemplary for the inertia term this reads as

$$ \begin{aligned} \int\limits_{0}^{T} {\varvec{\lambda}}^{T} {\mathbf{M\ddot{u}}}\;{\text{d}}t\; &= \,{\varvec{\lambda}} (T)^{T} {\mathbf{M\overline{{\dot{u}}}}} - {\varvec{\lambda}} (0)^{T} {\mathbf{M\dot{u}}}(0) \hfill \\ &\quad -{\varvec{\dot{\lambda}}}(T)^{T} {\mathbf{Mu}}(T) + {\varvec{\dot{\lambda}}}(0)^{T} {\mathbf{M\bar{u}}} \hfill \\ &\quad + \int\limits_{0}^{T} {\varvec{\ddot{\lambda}}^{T} {\mathbf{Mu}}\;{\text{d}}t.} \hfill \\ \end{aligned} $$

For the solvability of the adjoint final value problem, however, it must be assumed, that the initial values, such as initial displacement and initial velocity do not depend on the body shape.

We assume, that the external loads on a body are independent of the displacement. This currently excludes contact phenomena in the calculation. Further research from Fernandez et al. (2020) may be considered for contact between bodies.

The implicit derivative is supposed to vanish for any $\frac{\partial \mathbf{{u}}}{\partial r}= \varvec{\phi }$, which describes the adjoint equation of state in a time integral weak form

$$\begin{aligned} &\int \limits _0^T{\varvec{\lambda}}^T {{\textbf {K}}_{\textbf {T}}}\varvec{\phi } - \dot{\varvec{{\lambda}}}^T \mathbf{{C}}\varvec{\phi } + \ddot{\varvec{{\lambda}}}^T \mathbf{{M}}\varvec{\phi } \,\text {d}t\nonumber \\ & \qquad + \underbrace{\int \limits _0^T\left( \frac{\partial j}{\partial \mathbf{{u}}} - \frac{\partial }{\partial t} \frac{\partial j}{\partial \dot{\mathbf{{u}}}} + \frac{\partial ^2}{\partial t^2} \frac{\partial j}{\partial \ddot{\mathbf{{u}}}} \right) \varvec{\phi } \,\text {d}t}_{=:\left[ \mathcal {P}\right] _{(h)}} = 0\,, \end{aligned}$$

(15)

with the boundary condition ${\varvec{\lambda}} = \mathbf{{0}} \text{ on } {\varGamma }_D$ and the final values

$$\begin{aligned}&\left({\varvec{\lambda}}\left( T\right) ^T \mathbf{{M}}+ \left. \frac{\partial j}{\partial \ddot{\mathbf{{u}}}} \right| _{t= T} \right) \dot{\varvec{ {\phi }}} \left( T\right)&= 0 \, , \end{aligned}$$

(16)

$$\begin{aligned} &\left( - \dot{\varvec{{\lambda}}} \left( T\right) ^T \mathbf{{M}}+ {\varvec{\lambda}}\left( T\right) ^T \mathbf{{C}} + \left. \frac{\partial j}{\partial \dot{\mathbf{{u}}}} \right| _{t= T} - \left. \frac{\partial }{\partial t} \frac{\partial j}{\partial \ddot{\mathbf{{u}}}} \right| _{t= T} \right) \varvec{\phi }\left( T\right)&= 0 \, \end{aligned}$$

(17)

for all test functions $\varvec{\phi }$.

With the solution ${\varvec{\lambda}}$ of the final value problem (15)–(17), the topological derivative can be completed. In the material derivative (14) the implicit derivatives are eliminated and the explicit derivatives remain. The numerical evaluation of the explicit derivatives—and therefore of the topological derivative—is presented in Sect. 6.

Just as the displacement can be calculated at any point of the body at any time, the adjoint can also be calculated with the components in all spatial directions at any time. The implicit derivatives $\mathcal {P}$ of the functional are called pseudo loads, because they are independent from on the adjoint and therefore can be treated like a load vector. The pseudo loads establish the relationship between the functional and the adjoint in the first place. The lower index $\left[ \mathcal {P}\right] _{(h)}$ indicates the spatial discretization.

The final value problem is solved by executing the same time integration scheme, which was already used for the solution of the initial boundary value problem, backwards and from already existing time points with approximations on the time points lying temporally before them.

For the usage of the Newmark approximation for time integration, the approximation for the adjoint and for the adjoint velocity is first taken from (4) and (5) and resolved to ${\ddot{\varvec{{\lambda}}}^{(k-1)}}$ and ${\dot{\varvec{{\lambda}}}^{(k-1)}}$ for the solution of the final value problem. This yields the backward approximation of the adjoint acceleration and displacement

$$\begin{aligned} {\ddot{\varvec{{\lambda}}}^{(k-1)}}&= \alpha _1^\star \left( {\varvec{{\lambda}}}^{(k)} - {\varvec{{\lambda}}}^{(k-1)} \right) - \alpha _2^\star {\dot{\varvec{{\lambda}}}^{(k)}} - \alpha _3^\star {\ddot{\varvec{{\lambda}}}^{(k)}}\, , \end{aligned}$$

(18)

$$\begin{aligned} {\dot{\varvec{{\lambda}}}^{(k-1)}}&= \alpha _4^\star \left({\varvec{{\lambda}}}^{(k)} - {\varvec{{\lambda}}}^{(k-1)} \right) + \alpha _5^\star {\dot{\varvec{{\lambda}}}^{(k)}} + \alpha _6^\star {\ddot{\varvec{{\lambda}}}^{(k)}} \, , \end{aligned}$$

(19)

with $\beta ^*= 1+2\beta -2\gamma$ in the constants

$$\begin{aligned} \alpha _1^\star&= -\frac{2}{({\varDelta t^{(k)}})^2 \beta ^*}\, ,&\alpha _2^\star&=-\frac{2}{{\varDelta t^{(k)}}\beta ^*}\, , \nonumber \\ \alpha _3^\star&=-\frac{2(\beta -\gamma )}{\beta ^*}\, ,&\alpha _4^\star&=-\frac{2(\gamma -1)}{{\varDelta t^{(k)}} \beta ^*}\, , \nonumber \\ \alpha _5^\star&=-\frac{1-2\beta }{\beta ^*}\, ,&\alpha _6^\star&=-\frac{({\varDelta t^{(k)}})(2\beta -\gamma )}{\beta ^*}\, . \end{aligned}$$

The constants $\alpha _1^\star$, $\alpha _2^\star$, $\alpha _4^\star$ and $\alpha _6^\star$ depend on the time step size.

We discretize the pseudo-loads in time (upper index $\left[ \mathcal {P}\right] ^{(t)}$):

$$\begin{aligned}{} & {} \left[ \left[ \mathcal {P}\right] ^{(t)}\right] _{(h)} \nonumber \\{} & {} \quad = \sum _{k=0}^m\underbrace{ {w^{(k)}} \left( \frac{\partial j}{\partial \mathbf{{u}}^{(k)}} - \frac{\partial }{\partial t} \frac{\partial j}{\partial \dot{\mathbf{{u}}}^{(k)} } + \frac{\partial ^2}{\partial t^2} \frac{\partial j}{\partial \ddot{\mathbf{{u}}}^{(k)}} \right) \varvec{\phi }^{(k)} }_{=:\left( \left[ {P^{(k)}}\right] _{(h)}\right) ^T \varvec{\phi }^{(k)} } \,. \end{aligned}$$

(20)

With this discretization, the weak form of the adjoint equation of state (15) for every time point reads as

$$\begin{aligned}& {w}^{(k)} \left( {{\ddot{\varvec{{\lambda}}}}^{({k})^{T}}} \mathbf{{M}}- {{\dot{\varvec{{\lambda}}}}^{({k})^{T}}} \mathbf{{C}}+ {{\varvec{{\lambda}}}^{(k)}}^T {\mathbf{{K}}}_{\mathbf{{T}}}^{(k)} \right) \nonumber \\ &\qquad \qquad \qquad \qquad \quad \quad \;+ \left( \left[ {P^{(k)}}\right] _{(h)}\right) ^T =\mathbf{{0}} \,. \end{aligned}$$

(21)

Inserting the adjoint approximations (18) and  (19) and solving for the previous time step ${\varvec{{\lambda}}^{(k-1)}}$ leads to

$$\begin{aligned} &{w^{(k-1)}} \left( -\alpha _1^\star \mathbf{{M}} + \alpha _4^\star \mathbf{{C}}+ {\mathbf{{K}}}_{\mathbf{{T}}}^{(k-1)} \right) {\varvec{{\lambda }}}^{(k-1)}\\&\qquad \qquad \qquad \quad= - \left[ {P^{{(k - 1)}} } \right]_{{(h)}} \\&\qquad \qquad \qquad \qquad -\left( \alpha _1^\star \mathbf{{M}}- \alpha _4^\star \mathbf{{C}}\right){\varvec{\lambda }} ^{(k)}\\&\qquad \qquad \qquad \qquad -\left( -\alpha _2^\star \mathbf{{M}}- \alpha _5^\star \mathbf{{C}}\right) {\dot{\varvec{{\lambda }}}^{(k)}} \\&\qquad \qquad \qquad \qquad -\left( -\alpha _3^\star \mathbf{{M}}- \alpha _6^\star \mathbf{{C}}\right) {\ddot{\varvec{{\lambda }}}^{(k)}} \, . \end{aligned}$$

(22)

Starting with the final values (16) and (17) in the discretized form

$$\begin{aligned} \varvec{{\lambda}}^{(m)}&= - \mathbf{{M}}^{-1} \left( \frac{\partial j}{\partial \ddot{\mathbf{{u}}}^{(m)}} \right) ^T \, , \end{aligned}$$

(23)

$$\begin{aligned} {\dot{\varvec{{\lambda}}}^{(m)}}&= \mathbf{{M}}^{-1} \left(\vphantom{\left( \frac{\partial j}{\partial \dot{\mathbf{{u}}}^{(m)}} \right) ^T} \mathbf{{C}}\varvec{ {\lambda }}^{(m)} \right. \nonumber \left. + \left( \frac{\partial j}{\partial \dot{\mathbf{{u}}}^{(m)}} \right) ^T - \left( \frac{\partial }{\partial t} \frac{\partial j}{\partial \ddot{\mathbf{{u}}}^{(m)}} \right) ^T \right) \end{aligned}$$

(24)

and the final adjoint acceleration from the adjoint state equation (21) at the final time

$$\begin{aligned} {\ddot{\varvec{ {\lambda }}}^{(m)}}&= \mathbf{{M}}^{-1} \left( \mathbf{{C}}{\dot{\varvec{ {\lambda }}}}^{(m)} - {\mathbf{{K}}}_{\mathbf{{T}}}^{(m)} {}\varvec{ {\lambda }}^{(m)} - \left[ {P^{\,(m)}}\right] _{(h)} \right) \end{aligned}$$

(25)

the adjoint can be calculated sequentially from the already known adjoint ${\varvec{ {\lambda }}}^{(k)}$, adjoint velocity $\dot{\varvec{ {\lambda }}}{^{(k)}}$ and adjoint acceleration $\ddot{\varvec{ {\lambda }}} {^{(k)}}$. With the current adjoint ${\varvec{ {\lambda }}^{(k-1)}}$ inserted in (18) and  (19) all necessary quantities for the previous time step can be derived. The adjoint solution procedure is summarized in Algorithm 1.

For the trapezoidal time integration with the Newmark constants $\gamma =0.5$ and $\beta =0.25$, the set of matrices on the left in (22) is often referenced as effective stiffness matrix. If this matrix was already inverted for Newton’s algorithm for the solution of the initial boundary value problem, the inverted effective stiffness matrix can speed up the adjoint backward solution.

The discretization of the pseudo-loads depends on the functional $\mathcal {J}$ and its inner function $z$ in particular.

For the pseudo-load of the internal energy $\mathcal {J}_{\sigma }$, the time and space discretized form $j_{\sigma }$ from (8) is differentiated according to (20)

$$\begin{aligned} \left[ {P_\sigma ^{(k)}}\right] _{(h)}&= \left( \frac{\partial j_\sigma }{\partial {\mathbf{{u}}^{(k)}}}\right) ^T -\left( \frac{\partial }{\partial t}\frac{\partial j_\sigma }{\partial {\dot{\mathbf{{u}}}^{(k)}}} \right) ^T \\&= {w^{(k)}} \left( \mathbf{{K}}_{\mathbf{{T}}}^{(k)} \dot{\varvec{ {\mathbf{{u}}}}}^{(k)}_{} -\frac{\partial }{\partial t} \mathbf{{r}}\left( {\mathbf{{u}}^{(k)}}\right) \right) \\&= {w^{(k)}} \left( \mathbf{{K}}_{\mathbf{{T}}}^{(k)} \dot{\varvec{ {\mathbf{{u}}}}}^{(k)}_{} - \frac{\partial \mathbf{{r}}\left( {\mathbf{{u}}^{(k)}} \right) }{\partial {\mathbf{{u}}^{(k)}}} \frac{\partial {\mathbf{{u}}^{(k)}}}{\partial t} \right) \\&= {w^{(k)}} \left( \mathbf{{K}}_{\mathbf{{T}}}^{(k)} \dot{\varvec{ {\mathbf{{u}}}}}^{(k)}_{} - \mathbf{{K}}_{\mathbf{{T}}}^{(k)} \dot{\varvec{ {\mathbf{{u}}}}}^{(k)}_{} \right) \\&= \mathbf{{0}} \qquad \text {for } k=0, \ldots ,m \,. \end{aligned}$$

The pseudo load of the internal energy for the adjoint solution scheme “first differentiate then discretize” is equal to the zero vector for each time point. The time derivative of the vector of the internal forces $\frac{\partial }{\partial t} \mathbf{{r}}\left( {\mathbf{{u}}^{(k)}}\right)$ depends on the particular material modeling in the finite element solver. For the exemplary application in this paper and without further knowledge, we assume for simplification, that the vector of the internal forces is implicitly depending on the time via the displacement for the exemplary application in this paper. This assumption is equivalent to the assumption, that the time derivative of the 2nd Piola-Kirchhoff stress tensor $\varvec{S}$ is only implicitly depending on the time via the Green-Lagrange strain $\varvec{E}$ for the continuously defined functional $\mathcal {J}_{\sigma }$

$$\begin{aligned} \frac{\partial \varvec{S}}{\partial t}&= \left\langle \frac{\partial \varvec{S}}{\partial \varvec{E}} \, , \, \frac{\partial \varvec{E}}{\partial t} \right\rangle \\&= \left\langle \frac{\partial \varvec{S}}{\partial \varvec{E}} \, , \, \frac{1}{2}({{\,\text{Grad}\,}}\dot{\mathbf{{u}}}^{T}\varvec{F}+ \varvec{F}^T{{\,\text{Grad}\,}}\dot{\mathbf{{u}}}) \right\rangle \, . \end{aligned}$$

Then, the derivation of the inner function $z\left( \mathbf{{u}},\dot{\mathbf{{u}}}\right) := \varvec{F}\varvec{S} \cdot {{\,\text{Grad}\,}}\dot{\mathbf{{u}}}$ in (20) with respect to the displacement $\mathbf{{u}}$ and the velocity $\dot{\mathbf{{u}}}$ also leads to zero pseudo loads.

The final values are determined according to (23) and (25):

$$\begin{aligned} \varvec{ {\lambda }}^{(m)}_\sigma&= \mathbf{{0}} \,,\\ \dot{\varvec{ {\lambda }}}^{(m)}_\sigma&= \mathbf{{M}}^{-1} \mathbf{{r}}\left( {\mathbf{{u}}}_{}^{(m)} \right) \,, \\ \ddot{\varvec{ {\lambda }}}^{(m)}_\sigma&= \mathbf{{M}}^{-1} \left( \mathbf{{C}}\mathbf{{M}}^{-1} \mathbf{{r}}\left( {\mathbf{{u}}}_{}^{(m)} \right) \right) \, . \end{aligned}$$

The pseudo loads for the displacement functional $\mathcal {J}_u$ are the derivatives of (10) with respect to the displacement vector ${\mathbf{{u}}^{(k)}}$

$$\begin{aligned} \left[ {P_u^{(k)}}\right] _{(h)} = \frac{{w^{(k)}}}{T} \left( \frac{\mathbf{{l}}^T {\mathbf{{u}}^{(k)}} }{j_u} \right) ^{p-1} \mathbf{{l}} \, . \end{aligned}$$

(26)

At the time of maximum displacement, the pseudo load is also maximum. The larger the exponent p, the faster the pseudo loads become smaller.

The final values of the adjoint and adjoint velocity of the displacement functional are zero due to the independence from the velocity and acceleration:

$$\begin{aligned} \varvec{ {\lambda }}^{(m)}_u = {\dot{\varvec{ {\lambda }}}^{(m)}}_u = \mathbf{{0}}\, . \end{aligned}$$

The adjoint acceleration is determined using the pseudo load at the final time:

$$\begin{aligned} {\ddot{\varvec{ {\lambda }}}_u^{(m)}} =- \frac{{w^{(m)}}}{T}\left( \frac{\mathbf{{l}}^T {\mathbf{{u}}^{(m)}}}{j_u} \right) ^{p-1} \mathbf{{M}}^{-1} \mathbf{{l}} \, . \end{aligned}$$

Algorithm 1 Calculation of the adjoint “first differentiate then discretize”

5.2 First discretize then differentiate

One can also reverse the order and first consider the time discretized form and then differentiate. In contrast to the approach in the previous subsection, we follow the adjoint development approach used in the nonlinear sensitivity calculations with the density method presented in Ivarsson et al. (2018).

In addition to the discretization in the time domain of the extended functional $\hat{\mathcal {J}}$ from (13), the displacement and velocity approximations of the time integration (4) and (5) are added to the extended functional. Since time discretization is done first, followed by differentiation with respect to the design variables, this adjoint solution scheme is called “first discretize then differentiate”. The motivation behind this procedure is to determine the adjoint without the partial integration in the time domain. However, this is only possible if the relationship between the time points is established by the displacement and velocity approximations. Without these, no solution scheme would emerge.

The difficulty in adopting the approach lies in the formulation of the explicit derivations. In the density methods, the spatially discretized equilibrium equa- tions are added to the functional and the explicit derivatives are then calculated as derivatives of the system matrices. In the system matrices, however, no material derivative can take place. Therefore, the entire derivation must also be done in a spatially continuous form and only be spatially discretized for the elimination of the implicit derivative.

For the application of the adjoint method we extend the temporally discretized functional $\left[ \mathcal {J}\right] ^{(t)}$ both with the mechanical equilibrium integrally defined with respect to time, but also temporally discretized with the weighting factors ${w^{(k)}}$ (numerical quadrature) on the body with the cut-out ${\varOmega }_r$, and with the Newmark-approximations (4) and (5) with their own Lagrange multiplier $\varvec{ {\lambda }}_{\mathbf{{v}}}^{(k)}$ and $\varvec{ {\lambda }}_{\mathbf{{u}}}^{(k)}$:

$$\begin{aligned}&\left[ \hat{\mathcal {J}}\right] ^{(t)} = \nonumber \\&\left[ \mathcal {J}\right] ^{(t)} + \sum _{k=0}^m{w}^{(k)} \left( \int \limits _{{\varOmega }_r}\left( {{}^{u}{\varvec{F}}^{(k)}} {{}^{u}{\varvec{S}}^{(k)}} \cdot {{\,\text{Grad}\,}}{\varvec{ {\lambda }}^{(k)}} \right. \right. \nonumber \\&\qquad \qquad \qquad \qquad \quad \;+ c_d{ \dot{\mathbf{{u}}}^{(k)}} \cdot {\varvec{ {\lambda }}^{(k)}} \,\text {d}{\varOmega }+ \varrho {\ddot{\mathbf{{u}}}^{(k)}} \cdot {\varvec{ {\lambda }}^{(k)}} \nonumber \\&\qquad \qquad \qquad \qquad \quad \; \left. - \mathbf{{f}} \left( {t^{(k)}}\right) \cdot {\varvec{ {\lambda }}^{(k)}} \right) \,\text {d}{\varOmega }\nonumber \\&\qquad \qquad \qquad \qquad \quad \; \left. - \int \limits _{{\varGamma }_N} \varvec{ {\uptau }} \left( {t^{(k)}}\right) \cdot {\varvec{ {\lambda }}^{(k)}} \,\text {d}{\varGamma }\right) \nonumber \\ &\qquad \; + \sum _{k=1}^m {\varvec{ {\lambda }}_{\mathbf{{u}}}^{(k)}}^T \bigg ( {\mathbf{{u}}^{(k-1)}} + (\varDelta {t^{(k)}}) {\dot{\mathbf{{u}}}^{(k-1)}} \nonumber \\ &\qquad \qquad \; + \frac{(\varDelta {t^{(k)}})^2}{2}\left( (1-2 \beta ){\ddot{\mathbf{{u}}}^{(k-1)}} + 2\beta {\ddot{\mathbf{{u}}}^{(k)}}\right) - {\mathbf{{u}}^{(k)}} \bigg ) \nonumber \\ &\qquad \; + \sum _{k=1}^m {\varvec{ {\lambda }}_{\mathbf{{v}}}^{(k)}}^T \bigg ( {\dot{\mathbf{{u}}}^{(k-1)}} \nonumber \\ &\qquad \qquad\; +(\varDelta {t^{(k)}}) \left( (1-\gamma ){\ddot{\mathbf{{u}}}^{(k-1)}} + \gamma {\ddot{\mathbf{{u}}}^{(k)}}\right) - {\dot{\mathbf{{u}}}^{(k)}} \bigg ) \, . \end{aligned}$$

(27)

Again, the material derivation of $\left[ \hat{\mathcal {J}}\right] ^{(t)}$ leads to implicit and explicit derivatives. The explicit material derivative in this approach thus provides the time discretization of the explicit material derivative of the adjoint solution scheme “first differentiate then discretize” from (14). The implicit material derivatives of the extended time discretized functional in (27) are first spatially discretized individually. The infinitesimal section $\lim _{r\downarrow 0} c_r(\mathbf{{X}}_0)$ is again a null set, so the implicit derivatives can be defined on the entire body.

The implicit derivative terms are sorted by the derivatives of the displacement, the velocity and the acceleration with respect to the hole parameter $(\frac{\partial \mathbf{{u}}}{\partial r}, \frac{\partial \dot{\mathbf{{u}}}}{\partial r}$ and $\frac{\partial \ddot{\mathbf{{u}}}}{\partial r})$. Since for all implicit derivatives it is required to disappear, the sorted sums are set equal to zero, each for itself. The result is the adjoint solution scheme, where starting with the final values

$$\begin{aligned} w^{(m)}_{}\bigg (\mathbf{{M}}&+ \gamma \varDelta t^{(m)}_{} \mathbf{{C}}+ \beta \left( \varDelta t^{(m)}_{} \right) ^2\mathbf{{K}}_{\mathbf{{T}}}^{(m)} \bigg ) {\varvec{ {\lambda }}}^{(m)}_{} \nonumber \\&= - \bigg ( \beta \left( \varDelta {t^{(m)}}\right) ^2 \left( \frac{\partial j}{\partial {\mathbf{{u}}^{(m)}}}\right) ^T \nonumber \\&\qquad + \gamma \varDelta {t^{(m)}} \left( \frac{\partial j}{\partial {\dot{\mathbf{{u}}}^{(m)}}}\right) ^T + \left( \frac{\partial j}{\partial {\ddot{\mathbf{{u}}}^{(m)}}} \right) ^T \bigg ) \, , \end{aligned}$$

(28)

$$\begin{aligned} \varvec{ {\lambda }}_{\mathbf{{u}}}^{(m)}&= w^{(m)}_{} \mathbf{{K}}_{\mathbf{{T}}}^{(m)} {\varvec{ {\lambda }}}^{(m)}_{} + \left( \frac{\partial j}{\partial {\mathbf{{u}}^{(m)}}}\right) ^T \end{aligned}$$

(29)

and

$$\begin{aligned} \varvec{ {\lambda }}_{\mathbf{{v}}}^{(m)}&= {w}^{(m)} \mathbf{{C}}{\varvec{ {\lambda }}}^{(m)} + \left( \frac{\partial j}{\partial {\dot{\mathbf{{u}}}^{(m)}}}\right) ^T \end{aligned}$$

(30)

the solution of the equation

$$\begin{aligned}&{w}^{(k)}_{} \bigg (\mathbf{{M}}+ \gamma \varDelta {t}^{(k)}_{} \mathbf{{C}}+ \beta \left( \varDelta {t}^{(k)}_{} \right) ^2\mathbf{{K}}_{\mathbf{{T}}}^{(k)}\bigg ) \varvec{ {\lambda }}^{(k)}_{} \nonumber \\& \quad = - \beta \left( \varDelta {t^{(k)}}\right) ^2 \left( \frac{\partial j}{\partial {\mathbf{{u}}^{(k)}}}\right) ^T - \gamma \varDelta {t^{(k)}} \left( \frac{\partial j}{\partial {\dot{\mathbf{{u}}}^{(k)}}}\right) ^T \nonumber \\&\qquad - \left( \frac{\partial j}{\partial {\ddot{\mathbf{{u}}}^{(k)}}}\right) ^T \nonumber \\&\qquad - \varvec{ {\lambda }}_{\mathbf{{u}}}^{(k+1)}\bigg (\frac{1}{2} \left( \varDelta {t^{(k+1)}}\right) ^2 - \beta \left( \varDelta {t^{(k+1)}}\right) ^2 \nonumber \\&\qquad \qquad \qquad \qquad + \beta \left( \varDelta {t^{(k)}}\right) ^2 + \gamma \varDelta {t^{(k)}}\varDelta {t^{(k+1)}}\bigg )\nonumber \\&\qquad - \varvec{ {\lambda }}_{\mathbf{{v}}}^{(k+1)}\bigg (\varDelta {t^{(k+1)}} - \gamma \varDelta {t^{(k+1)}} + \gamma \varDelta {t^{(k)}}\bigg ) \end{aligned}$$

(31)

gives the adjoint $\varvec{ {\lambda }}^{(k)}$. The Lagrange multipliers of the displacement approximation and the velocity approximation for each time point k are calculated through

$$\begin{aligned} \varvec{ {\lambda }}_{\mathbf{{u}}}^{(k)}&= \left( \frac{\partial j}{\partial {\mathbf{{u}}^{(k)}}}\right) ^T + \varvec{ {\lambda }}_{\mathbf{{u}}}^{(k+1)} +{w}^{(k)}_{} \mathbf{{K}}_{\mathbf{{T}}}^{(k)} \varvec{ {\lambda }}^{(k)}_{}\, . \end{aligned}$$

(32)

$$\begin{aligned} \varvec{ {\lambda }}_{\mathbf{{v}}}^{(k)}&= \varvec{ {\lambda }}_{\mathbf{{v}}}^{(k+1)} + \varDelta {t^{(k+1)}} \varvec{ {\lambda }}_{\mathbf{{u}}}^{(k+1)} \nonumber + {w}^{(k)} \mathbf{{C}}{\varvec{ {\lambda }}^{(k)}} + \left( \frac{\partial j}{\partial {\dot{\mathbf{{u}}}^{(k)}}}\right) ^T , \end{aligned}$$

(33)

For the initial value remains

$$\begin{aligned} {\varvec{ {\lambda }}^{(0)}} = {w^{(0)}}^{-1} \mathbf{{M}}^{-1} \bigg ( - \left( \frac{\partial j}{\partial {\ddot{\mathbf{{u}}}^{(0)}}}\right) ^T& - \left( 1-\gamma \right) \varDelta {t^{(1)}} \varvec{ {\lambda }}_{\mathbf{{v}}}^{(1)} \nonumber \\& - \frac{1-2\beta }{2} \left( \varDelta {t^{(1)}}\right) ^2\varvec{ {\lambda }}_{\mathbf{{u}}}^{(1)} \bigg ) . \end{aligned}$$

(34)

If the backward solution rule (31) is multiplied by $(\beta (\varDelta {t^{(k)}})^2)^{-1}$ one again discovers the effective stiffness matrix from the solution of the initial boundary value problem on the left-hand side, as in (21).

The pseudo loads for the internal energy are given by

$$\begin{aligned} \left( \frac{\partial j_{\sigma }}{\partial {\mathbf{{u}}^{(k)}}}\right) ^T =&\quad{w^{(k)}} \mathbf{{K}}_{\mathbf{{T}}}^{(k)} \dot{\varvec{ {\mathbf{{u}}}}}^{(k)}_{} \, , \\ \left( \frac{\partial j_{\sigma }}{\partial {\dot{\mathbf{{u}}}^{(k)}}}\right) ^T =&\quad{w^{(k)}} \mathbf{{r}}\left( {\mathbf{{u}}^{(k)}}\right) \, , \end{aligned}$$

whereas the pseudo load for the displacement functional does not differ from the pseudo loads in “first differentiate then discretize” in (26).

Both adjoint procedures look similar at first, but they cannot be transformed into each other. The solutions are also not the same as one can see by comparing the final values in (23) and (28). But they “look” very similar (as far as one could visualize the non-physical adjoints) and give similar results. This is a well-known but still unsolved problem of transient sensitivity analysis. For this purpose, a spring-mass oscillator minimal example exists in Jensen et al. (2014). This issue is also addressed in Fernandez and Tortorelli (2018). In Weider (2021) the differences of the schemes are worked up in more detail.

What both procedures have in common is the linearity in each time point. So in each time a linear system of equations is solved, which is calculated with the temporary tangent stiffness. If sensitivities for several functionals are to be determined within one load scenario, the pseudo loads of all functionals can be stacked next to each other and, due to the linearity of the adjoint solution schemes, the adjoints for all loadcases can be determined in parallel with the costs of one adjoint backward calculation. However, if several crash loadcases are involved, a separate adjoint backward calculation must be performed for each loadcase, since the tangential stiffness matrices differ in the loadcases due to different deformation and stress states.

Algorithm 2 Calculation of the adjoint according to “first discretize then differentiate”

6 Calculation of the topological derivative

With the solution $\mathbf{{u}}$ and its derivatives with respect to the time, the adjoint solution $\varvec{ {\lambda }}$ and the explicit material derivative $\left( \tfrac{\partial \mathcal {J}}{\partial r}\right) _E$ the topological derivative at the point $\mathbf{{X}}_0$ remains as

$$\begin{aligned} \mathcal {T}\!\mathcal {J}\left( \mathbf{{X}}_0\right)&= \lim _{r\downarrow 0} \left( \frac{\,\text {d}|c_r(\mathbf{{X}}_0)|}{\,\text {d}r}\right) ^{-1} \Bigg ( \left( \frac{\partial \mathcal {J}}{\partial r}\right) _E \Bigg . \nonumber \\&\qquad \quad \Bigg . - \int \limits _0^T\int \limits _{{\varGamma }_r}\varvec{F}\varvec{S} \cdot \nabla \varvec{ {\lambda }} - c_d\frac{\partial \mathbf{{u}}}{\partial t} \cdot \varvec{ {\lambda }} \Bigg . \nonumber \\&\qquad \quad \Bigg . + \varrho \frac{\partial ^2 \mathbf{{u}}}{\partial t^2} \cdot \varvec{ {\lambda }} - \mathbf{{f}}(t) \cdot \varvec{ {\lambda }} \,\text {d}{\varGamma }\,\text {d}t\Bigg ) \, . \end{aligned}$$

(35)

The explicit material derivative for the internal energy from (7) is given through the well-known integral on the boundary of the hole

$$\begin{aligned} \left( \frac{\partial \mathcal {J}_{\sigma }}{\partial r}\right) _E = - \int \limits _0^T\int \limits _{{\varGamma }_r}\varvec{F}\varvec{S} \cdot {{\,\text{Grad}\,}}\dot{\mathbf{{u}}} \,\text {d}{\varGamma }\,\text {d}t\, . \end{aligned}$$

The single point displacement functional in (9) is not explicitly depending on the body ${\varOmega }$ and therefore admits no material derivative with respect to the parameter $r$:

$$\begin{aligned} \left( \frac{\partial \mathcal {J}_u}{\partial r}\right) _E = 0 \, . \end{aligned}$$

The integrals on the boundary of the hole admit analytical solutions only for the pure linear case. That is, for small displacements and rotations, small strains and linear elasticity the integrals are evaluated with the use of Airy’s stress function (Eschenauer et al. 1994) and the complex potentials approach (Schumacher 1995). The underlying conditions considered in this paper are the contrary case: large displacements and rotations, large strains and material nonlinearity, for which no analytical stress solutions are known. Thus, there is a need for an approximation of all values on the boundary of the hole.

In the following, we present a material interpolation reflecting the temporary material behavior and discuss the evaluation of the boundary integrals in particular using the shell formulation with a cylindrical cut-out. However, the calculation can in principle also be adapted for solid models.

6.1 Material interpolation for plastic strain

With the intention of approximating the linear formulas for the circumferential stress coming from Airy’s stress function e. g. with modified material parameters also for nonlinear material behavior, we develop a phenomenological material interpolation in this section. The basic idea is: At each time point ${t^{(k)}}$, a temporary linear material behavior is present due to the linearized solution using implicit time integration with the tangent stiffness $\mathbf{{K}}_{\mathbf{{T}}}^{(k)}$, which is reproduced with the material interpolation.

The temporary material in the tangential stiffness matrix $\mathbf{{K}}_{\mathbf{{T}}}^{(k)}$ is characterized by the tangent modu- lus $E_{\text {T}}$ and an effective Poisson’s ratio $\nu _{\text {eff}} \in \left[ \nu , 0.5\right]$.

Poisson’s ratio $\nu$ is given only for the elastic region of a material. Since plastic strains are constant in volume, the Poisson’s ratio for pure plastic strain is equal to 0.5. For plastic strain, the plastic part dominates the total strain very quickly, since the elastic strain range is very small for metallic materials. Thus, as plastic strain increases, the Poisson’s ratio $\nu _{\text {eff}}$ must increase from the elastic $\nu$ to the plastic 0.5.

The effective plastic strain $\varepsilon _{\text {eff}}^{pl}$ is incrementally summed up during the material calculation in the finite element calculation program and is available as uniaxial comparative strain for each integration point. With a continuously differentiable step function with the constant $s_1$ for the slope, the constant $s_2$ for the shift along the strain axis and the yield strain $\varepsilon _{\text {eff}}^e = \frac{\sigma _y}{E}$, a smooth transition to the plastic Poisson’s ratio is achieved

$$\begin{aligned}{} & {}\nu_{\text {eff}}\left(\varepsilon_{\text {eff}}^{pl}\right)\! \!\!\! &&:= \frac{0.5 + \nu }{2} \nonumber \\{} & {} &&+ \frac{0.5 - \nu }{2}\tanh \left( s_1 \left( \varepsilon _{\text {eff}}^{pl} - s_2 \, \varepsilon _{\text {eff}}^e \right) \right) . \end{aligned}$$

(36)

For the examples in Sect. 7 the constants $s=10$ and $s_2=75$ are chosen. The graph of the effective Poisson’s ratio is depicted in Fig. 3a.

Fig. 3

Material interpolation depending on the effective plastic strain

Another abbreviation is introduced for the factor that frequently occurs in moment calculations:

$$\begin{aligned} \nu ^\prime := \frac{2\left( 1+\nu _{\text {eff}}\right) }{3+\nu _{\text {eff}}} \, . \end{aligned}$$

The strain hardening as a function of effective plastic strain is given as a piecewise linear function in the aluminum material model used in this work.² Pairs of values ($\varepsilon _{\text {eff}}^{pl}$, $\tilde{\sigma }_y$) of the effective plastic strain and the effective stress are captured in Table 1 and Fig. 3b. All other material parameters used are summarized in Table 2. During yielding, the effective stress is equal to the temporary local yield stress $\tilde{\sigma }_y$. Linear interpolation is performed between the pairs of values. The plastic modulus $E_{p}$ is taken as the gradient of this curve. It is, therefore, constant between two interpolation points:

$$\begin{aligned} E_{p}= \frac{\varDelta \tilde{\sigma }_y\left( \varepsilon _{\text {eff}}^{pl}\right) }{\varDelta \varepsilon _{\text {eff}}^{pl}}\, . \end{aligned}$$

The tangent modulus $E_{\text {T}}$ of the stress–strain curve reflects the local gradient of the stress with respect to the total strain. The plastic modulus $E_{p}$ is the gradient of the stress with respect to the effective plastic strain as in shown in Fig. 3b. According to Bathe (2014), the tangent modulus $E_{\text {T}}$ can be calculated from the plastic modulus as

$$\begin{aligned} E_{\text {T}}= \frac{E_{p}E}{E_{p}+ E}\, . \end{aligned}$$

(37)

This tangent modulus is only used when plastic strain occurs. Without plastic strain, the modulus of elasticity $E$ is used, so that finally the effective tangent stiffness $E_{p}$ shown in Fig. 3c results.

Table 1 Yield stress over effective plastic strain

Table 2 Elastic parameters of the considered aluminum

The material interpolation developed here is understood as the interpolation of material properties under plastic deformation and should not be mixed up with material interpolations in density methods, where the material properties are transferred to an artificial material with lower density.

Since the actual material calculation happens in the finite element program without the user seeing the data, this interpolation is based only on the phenomenological observations. With more detailed knowledge of the material calculation, the material interpolation can be improved significantly.

6.2 Calculation for shell elements

First, we introduce the basic shell description and then give the particular formulas for the individual integral terms of the topological derivatives.

6.2.1 Basic shell description

Curved structural members that have a small extension in thickness direction compared to their planar sizes, are called shells. These are, for example, sheets metals. In order to represent the complex geometry e. g. of car bodies, shell elements are used in the calculation. The most important features of the 5-parameter theory will be summarized in the following.

The classical shell description is based on the definition of the midsurface $\mathcal {M}$ and the shell thickness $h$. The surface parameters are usually chosen as convective coordinates $\xi _{\alpha }$, where the Greek Index $\alpha$ takes the values 1 and 2. The parameterization and the geometry are shown in Fig. 4. To start with, we can determine the position vector $\mathbf{{X}}$ of a point in the shell continuum in the initial configuration by

$$\begin{aligned} \mathbf{{X}}\left( \xi _1, \xi _2, \xi _3, t\right) = \mathbf{{X}}_{\mathcal {M}} \left( \xi _{\alpha }, t\right) + \frac{h}{2} \xi _3\mathbf{{N}}\left( \xi _{\alpha }, t\right) \, , \end{aligned}$$

(38)

where $\mathbf{{N}}$ is the normal unit vector to the midsurface $\mathcal {M}$ in the initial configuration. The thickness parameterization by $\frac{h}{2}\xi _3$ with $\xi _3\in \left[ -1, 1\right]$ is chosen according to the isoparametric formulation of the discretization.

Fig. 4

Covariant basis vectors of the shell continuum in the reference (left) and the current configuration (right), modified from Wriggers (2008)

With the assumption that plane cross sections remain plane, the position vector in the current configuration may be described by

$$\begin{aligned} \mathbf{{x}}\left( \xi _1, \xi _2, \xi _3, t\right) = \mathbf{{x}}_{\mathcal {M}}\left( \xi _{\alpha }, t\right) + \frac{h}{2} \xi _3\, \mathbf{{d}}\left( \xi _{\alpha }, t\right) \, , \end{aligned}$$

(39)

with $\xi _3\in \left[ -1, 1\right]$. The translational displacement is the difference vector of the midsurface location in the reference and the current configuration:

$$\begin{aligned} \mathbf{{u}}\left( \xi _{\alpha }, t\right) = \mathbf{{x}}_{\mathcal {M}}\left( \xi _{\alpha }, t\right) - \mathbf{{X}}_{\mathcal {M}}\left( \xi _{\alpha }, t\right) \, . \end{aligned}$$

(40)

The director $\mathbf{{d}}$ describes the distortion of the cross section caused by the deformation. According to the Reissner-Mindlin plate theory, the director differs from the surface normal vector in the current configuration. For the elementary and simplified concept of the 5-parameter theory, no strain in thickness direction is allowed. Hence, the length of the director remains constant $||\mathbf{{d}}||_2 =1$. This constraint can be described as follows: The director follows from a pure rotation with the matrix $\mathbf{{R}}$ from the initial normal vector

$$\begin{aligned} \mathbf{{d}}= \mathbf{{R}}\mathbf{{N}}\, . \end{aligned}$$

The rotation matrix of the director $\mathbf{{R}}$ is represented by the rotation axis $\varvec{ {\theta }}/ ||\varvec{ {\theta }}||_2$ from the three rotational degrees of freedom around the three global axes $\varvec{ {\theta }}= \left( \theta _1, \theta _2, \theta _3\right) ^T$ and the rotation angle $\theta = ||\varvec{ {\theta }}||_2$ with the so-called Rodrigues formula

$$\begin{aligned} \mathbf{{R}}\left( t\right) = \varvec{1}+ \frac{\sin \theta \left( t\right) }{\theta \left( t\right) } [\varvec{ {\theta }}\left( t\right) ]_{\textsf {x}} + \frac{1-\cos \theta \left( t\right) }{\theta \left( t\right) ^2} [\varvec{ {\theta }}\left( t\right) ]_{\textsf {x}}^2\, , \end{aligned}$$

(41)

with the cross-product matrix defined as follows:

$$\begin{aligned}{}[\,\varvec{ {\theta }}\,]_{\textsf {x}} := \begin{pmatrix} 0 &{} - \theta _3 &{} \theta _2 \\ \theta _3 &{} 0 &{} -\theta _1 \\ -\theta _2 &{} \theta _1 &{} 0 \\ \end{pmatrix}\, . \end{aligned}$$

To shorten the notation, the derivatives with respect to the convective coordinates are represented by the lower comma: $\frac{\partial \,\square }{\partial \xi _{\alpha }} =: {\square }_{,\alpha }$. The covariant basis vectors of the initial configuration are the tangents with respect to the convective coordinates and are represented by

$$\begin{aligned} \mathbf{{G}}_{\alpha }&= \frac{\partial \mathbf{{X}}}{\partial \xi _{\alpha }} = \mathbf{{A}}_{\alpha } + \xi _3\frac{h}{2} {\mathbf{{N}}}_{,\alpha } \, ,\\ \mathbf{{G}}_{3}&= \frac{\partial \mathbf{{X}}}{\partial \xi _3} = \frac{h}{2} \mathbf{{N}}\, , \end{aligned}$$

with the tangent vectors $\mathbf{{A}}_\alpha = \frac{\partial \mathbf{{X}}_\mathcal {M}}{\partial \xi _{\alpha }}$ on the midsurface. One also needs these tangent vectors for the determination of the midsurface normal vector

$$\begin{aligned} \mathbf{{N}}= \frac{\mathbf{{A}}_1 \times \mathbf{{A}}_2}{||\mathbf{{A}}_1 \times \mathbf{{A}}_2 ||_2} \end{aligned}$$

and its derivative

$$\begin{aligned} {\mathbf{{N}}}_{,\alpha } &= \frac{{\mathbf{{A}}_1}_{,\alpha }\times \mathbf{{A}}_2 + \mathbf{{A}}_1 \times {\mathbf{{A}}_2}_{,\alpha }}{||\mathbf{{A}}_1 \times \mathbf{{A}}_2||_2} \\ &-\frac{\mathbf{{N}}}{||\mathbf{{A}}_1 \times \mathbf{{A}}_2||_2^2} \left( ||\mathbf{{A}}_2||^2_2 \mathbf{{A}}_1 \cdot {\mathbf{{A}}_1}_{,\alpha } + ||\mathbf{{A}}_1||^2_2 \mathbf{{A}}_2 \cdot {\mathbf{{A}}_2}_{,\alpha } \right. \\& \qquad- \left. \left( \mathbf{{A}}_1 \cdot \mathbf{{A}}_2 \right) \left( {\mathbf{{A}}_1}_{,\alpha } \cdot \mathbf{{A}}_2 + \mathbf{{A}}_1 \cdot {\mathbf{{A}}_2}_{,\alpha } \right) \right) \,. \end{aligned}$$

The related contravariant basis vectors are computed from the covariant basis vectors via the orthogonality relation:

$$\begin{aligned} \mathbf{{G}}_{i}^{}\cdot \mathbf{{G}}^{k}_{} = \delta _i^k \qquad \text {with} \qquad \delta _i^k = \left\{ \begin{array}{ll} 1, &{}\text {if } i=k,\\ 0, &{}\text {else}\, . \end{array} \right. \end{aligned}$$

Here, it is suitable to define a local orthonormal system $\mathcal {E}^{\text {loc}}$ in the midsurface that serves as reference system for the combined integration of the state and the adjoint on the hole boundary. Starting from the tangent vector at the shell midsurface $\mathbf{{A}}_1$ and the shell normal vector $\mathbf{{N}}$, we define the unit vectors

$$\begin{aligned} \mathbf{{e}}_1^{\text {loc}} = \frac{\mathbf{{A}}_1}{||\mathbf{{A}}_1||_2} \qquad \text {und} \qquad \mathbf{{e}}_2^{\text {loc}} = \mathbf{{N}}\times \mathbf{{e}}_1^{\text {loc}} \end{aligned}$$

as shown in Fig. 5. The basis system forms in the sequence

$$\begin{aligned} \mathcal {E}^{\text {loc}} = \left\{ \mathbf{{e}}_1^{\text {loc}}, \mathbf{{e}}_2^{\text {loc}}, \mathbf{{N}} \right\} \end{aligned}$$

(42)

an orthonormal system.

Fig. 5

Definition of the local orthonormal system in the midsurface

For the calculation of other kinematic quantities, one also needs the covariant basis vectors in the current configuration. Therefore, the tangent vectors on the midsurface in the current configuration are calculated in the same manner as the tangent vectors in the initial configuration:

$$\begin{aligned} \mathbf{{a}}_{\alpha } = \frac{\partial \mathbf{{x}}_{\mathcal {M}}}{\partial \xi _{\alpha }} \, . \end{aligned}$$

(43)

The derivative of the director with respect to the convective coordinates is calculated by the product rule

$$\begin{aligned} {\mathbf{{d}}}_{,\alpha } = \mathbf{{R}}{\mathbf{{N}}}_{,\alpha } + {\mathbf{{R}}}_{,\alpha } \mathbf{{N}}\, , \end{aligned}$$

(44)

where the derivative of the rotation is given by the chain rule

$$\begin{aligned} {\mathbf{{R}}}_{,\alpha } &= \frac{\theta \cos \theta - \sin \theta }{\theta ^3} \varvec{ {\theta }} \cdot {\varvec{ {\theta }}}_{,\alpha } [\, \varvec{ {\theta }}\,]_{\textsf {x}} + \frac{\sin \theta }{\theta }[{\varvec{ {\theta }}}_{,\alpha }]_{\textsf {x}} \nonumber \\ &+ \frac{2 \cos \theta -2 +\theta \sin \theta }{\theta ^3} \varvec{ {\theta }} \cdot {\varvec{ {\theta }}}_{,\alpha } [\,\varvec{ {\theta }}\,]_{\textsf {x}}^2 \nonumber \\ & \quad+ \frac{1-\cos \theta }{\theta ^2}\left( [\, \varvec{ {\theta }}\, ]_{\textsf {x}} [{\varvec{ {\theta }}}_{,\alpha }]_{\textsf {x}} + [{\varvec{ {\theta }}}_{,\alpha }]_{\textsf {x}}[\,\varvec{ {\theta }}\,]_{\textsf {x}} \right) \,. \end{aligned}$$

(45)

With these derivatives, we can fully describe the covariant basis vectors with respect to the current configuration

$$\begin{aligned} \mathbf{{g}}_{\alpha }&= \frac{\partial \mathbf{{x}}}{\partial \xi _{\alpha }} = \mathbf{{a}}_\alpha +\frac{h}{2} \xi _3{\mathbf{{d}}}_{,\alpha } \, , \end{aligned}$$

(46)

$$\begin{aligned} \mathbf{{g}}_{3}&= \frac{\partial \mathbf{{x}}}{\partial \xi _3} = \frac{h}{2} \mathbf{{d}}\, . \end{aligned}$$

(47)

In practical usage, the derivatives with respect to the convective coordinates are calculated using the derivatives of the finite element ansatz functions with respect to the isoparametric coordinates.

6.2.2 Calculation model of the explicit derivative of the mixed adjoint energy

With the material interpolation from Sect. 6.1 the individual integral terms of the topological derivative (35) can be specified especially for the previously described shell formulation. In this case, the cut-out is a cylindrical hole in the shell.

First, we consider the derivative of the mixed adjoint energy

$$\begin{aligned} - \left( \frac{\,\text {d}|c_r(\mathbf{{X}}_0)|}{\,\text {d}r}\right) ^{-1} \int \limits _0^T\int \limits _{{\varGamma }_r}{}^{u}{\varvec{F}} {}^{u}{\varvec{S}} \cdot {{\,\text{Grad}\,}}{\varvec{\lambda}} \,\text {d}{\varGamma }\,\text {d}t\, . \end{aligned}$$

This depends on one hand on the deformation gradient ${}^{u}{\varvec{F}}$ and the 2nd Piola-Kirchhoff stress tensor ${}^{u}{\varvec{S}}$ of the solution of the initial boundary value problem, and on the other hand on the gradient of the adjoint ${{\,\text{Grad}\,}}{\varvec{\lambda}}$. The superscript ${}^{u}$ on the left emphasizes the explicit dependency on the state solution. At each time and at each point on the hole boundary, quantities from the load calculation and the adjoint calculation are multiplied. Then the product is integrated on the hole boundary and the hole boundary integral is integrated in time.

Available results from the finite element simulation of the initial boundary value problem are true stress and strain tensors in every integration point, translational and rotational displacements, velocities and accelerations in every node and all values are available in every time step. Also the adjoint translational and rotational displacements can be derived by one of the adjoint schemes from Sect. 5. From the nodal values, the translational and rotational quantities within the body can be deduced with the help of the ansatz functions.

For the point $\mathbf{{X}}_0$ and the surrounding, the state and the adjoint are known. We consider the infinitesimal cylindrical hole around the point $\mathbf{{X}}_0$ in the shell. The insertion of this hole is a perturbation of the homogeneous stress state in the shell section, which is governed by the section forces $f_{xx}$, $f_{xy}$, and $f_{yy}$ and moments $m_{xx}$, $m_{xy}$, and $m_{yy}$ in Fig. 6. The surrounding cut-out is thereby very large compared to the dimensioning parameter $r$. We call this cut-out microcell because it represents the mechanical behavior at the microscopic level. All quantities of the microcell are related to the local basis system $\mathcal {E}^{\text {loc}}$, whose construction is given by the covariant basis vectors in (42).

Fig. 6

Microcell with cylindrical hole defined by cross-sectional forces and moments

The state stresses and deformation gradient have to be determined by the cross-sectional forces and moments from the solution of the initial boundary value problem. The adjoint gradient is specified by the corresponding forces and moments from the adjoint solution (depicted with superscript ^λ). We emphasize here, that these are not necessarily of the dimension of a force. Figure 7 shows the decomposition into the state and adjoint solutions and the respective resulting internal forces on the microcell.

Fig. 7

Decomposition of the derivative of the mixed adjoint energy into state and adjoint solution

If we relate the boundary integral to the current configuration of the initial boundary value problem, the scalar product property and the symmetric Cauchy stress tensor can be used to formulate the derivative of the mixed adjoint energy on the hole boundary that is deformed by the displacement $\mathbf{{u}}$:

$$\int \limits _{{\varGamma }_r}{}^{u}{\varvec{F}} {}^{u}{\varvec{S}} \cdot {{\,\text{Grad}\,}}{\varvec{\lambda}} \,\text {d}{\varGamma }= \int \limits _{{{\varGamma }_r}\left( t\right) } {}^{u}{\varvec{\sigma }} \cdot {}^{\lambda } {\varvec{\varepsilon}} \,\text {d}{\varGamma }\, .$$

From the solution of the adjoint final value problem, the gradient of the adjoint, respectively the adjoint infinitesimal strain tensor

$$\begin{aligned} {}^{\varvec{{\lambda}}}{\varvec\varepsilon} = {\varvec{\varepsilon}}\left(\varvec{{\lambda}}\right) = \frac{1}{2}\left( {{\,\text{Grad}\,}}{\varvec{\lambda}} + {{\,\text{Grad}\,}}{\varvec{\lambda}}^T\right) \end{aligned}$$

is needed. And from the state solution a deeper knowledge about the deformation of the hole boundary in form of the displacement gradient or about the displacements on the boundary respectively is necessary. For the Cauchy stress estimation, the second description in the current configuration seems easier to approximate, since analytical formulas for the stresses exist in the linear case. In addition to the stress and adjoint strain tensor one needs in particular a prediction for the deformation of the boundary. So, both integral formulations have the same pitfalls and need a simplification.

We therefore assume, that the hole is sufficiently small in contrast to the microcell, so the deformation of the boundary can be neglected for the calculation of the boundary integrals. The integration of the derivative of the mixed adjoint energy and further on also of the explicit derivative of the internal energy is performed with the transformation formula for the integration on submanifolds on the undeformed cylinder section

$$\begin{aligned} \int \limits _{{{\varGamma }_r}\left( t\right) } {}^{u}{\varvec{\sigma }} \cdot {}^{\varvec{ {\lambda }}}{\varvec\varepsilon} \,\text {d}{\varGamma }\approx \int \limits _{-\frac{h}{2}}^{\frac{h}{2}} \int \limits _0^{2\pi } {}^{u}{\varvec{\sigma }} \cdot {}^{\varvec{ {\lambda }}}{\varvec\varepsilon} \, r\,\text {d}\varphi \,\text {d}z\, . \end{aligned}$$

(48)

This aspect is also taken up again when calculating the derivative of the inertia term. With Cesáro’s formula (1906), this simplifying assumption can possibly be corrected in future investigations, when the shape of the hole is determined by the path integration of the strains.

The combined integration of the stresses from the initial boundary value problem with the adjoint linearized strain tensor is done in the tangential coordinate system $\mathcal {E}^{\text {loc}}$ in the shell midsurface defined in (42). For both the displacement $\mathbf{{u}}$ and the adjoint ${\varvec{{\lambda}}}$, this coordinate system is no longer tangential to the midsuface in the particular current configuration. Since the adjoint and the displacement do not need to have any similarities, the location of the tangent vector $\mathbf{{A}}_1$, from which the vector $\mathbf{{e}}_1^{\text {loc}}$ is determined, is used as the reference location for the integration variable in peripheral direction $\varphi$. This establishes a common parameterization with the angular position $\varphi =0$ on the tangent vector. In Fig. 8 is outlined how the tangent vector $\mathbf{{A}}_1$ is transformed into the current configurations ${}^{u}{\mathbf{{a}}}_1$ and ${}^{\lambda}{\mathbf{{a}}}_1$ where it is used to construct the basis systems. The basis systems are time and location dependent and must be computed for both the displacement and the adjoint.

Fig. 8

Common reference coordinate system $\mathcal {E}^{\text {loc}}$ for state stresses and adjoint strain tensors

The tangent vectors in the current configuration of the initial boundary value problem from (43) are denoted by

$$\begin{aligned} {}^{u}{\mathbf{{e}}}_1^{\text {loc}} =&\frac{{}^{u}{\mathbf{{a}}}_1}{||{}^{u}{\mathbf{{a}}}_1||_2}\,, \quad {}^{u}{\mathbf{{n}}} = \frac{{}^{u}{\mathbf{{a}}}_1 \times {}^{u}{\mathbf{{a}}}_2}{||{}^{u}{\mathbf{{a}}}_1 \times {}^{u}{\mathbf{{a}}}_2||_2} \end{aligned}$$

and

$$\begin{aligned} {}^{u}{\mathbf{{e}}}_2^{\text {loc}} =&{}^{u}{\mathbf{{n}}} \times {}^{u}{\mathbf{{e}}}_1^{\text {loc}}\, . \end{aligned}$$

The transformation of matrices and tensors from the global cartesian coordinate system to the local orthonormal system is done with the matrix ${}^{u}{\mathbf{{E}}}{^{\text {loc}}}$, which is composed of the basis vectors:

$$\begin{aligned} {}^{u}{\mathbf{{E}}}{^{\text {loc}}} = \left( {}^{u}{\mathbf{{e}}}_1^{\text {loc}}, {}^{u}{\mathbf{{e}}}_2^{\text {loc}}, {}^{u}{\mathbf{{n}}}\right) \, . \end{aligned}$$

For the adjoint basis system, the tangential vectors are the derivatives of the adjoint map**

$$\begin{aligned} {}^{\lambda}{\mathbf{{x}}_{\mathcal {M}}{(\mathbf{{X}}, t)}} = \mathbf{{X}}_{\mathcal {M}} +{\varvec{\lambda}}(\mathbf{{X}}, t) \, . \end{aligned}$$

(49)

Similar to the tangent vector calculation for shells in the current state configuration (43) the adjoint tangent vectors are calculated for the current adjoint configuration by

$$\begin{aligned} \mathbf{{a}}_{\alpha } = \frac{\partial {}^{\lambda }{\mathbf{{x}}} _{\mathcal {M}}}{\partial \xi _{\alpha }} \, . \end{aligned}$$

Equivalently, the basis system for the adjoint is given by the adjoint basis vectors

$$\begin{aligned} {}^{\lambda}{\mathbf{{e}}}_1^{\text {loc}} =&\frac{{}^{\lambda}{\mathbf{{a}}}_1}{||{}^{\lambda}{\mathbf{{a}}}_1||_2}\,, \quad ^{\lambda}\mathbf{{n}}= \frac{^{\lambda}{\mathbf{{a}}}_1 \times {}^{\lambda}{\mathbf{{a}} }_2}{||^{\lambda}{\mathbf{{a}}}_1 \times {}^{\lambda}{\mathbf{{a}}}_2||_2} \end{aligned}$$

and

$$\begin{aligned} {}^{\lambda}{\mathbf{{e}}}_2^{\text {loc}} =&{}^{\lambda}{\mathbf{{n}}} \times {}^{\lambda}{\mathbf{{e}}}_1^{\text {loc}} \, . \end{aligned}$$

The transformation matrix for the adjoint basis system is accordingly the 3-tuple from the adjoint basis vectors

$$\begin{aligned} {}^{\lambda}{\mathbf{{E}}}{^{\text {loc}}} = \left( {}^{\lambda}{\mathbf{{e}}}_1^{\text {loc}}, {}^{\lambda}{\mathbf{{e}}}_2^{\text {loc}}, {}^{\lambda}{\mathbf{{n}}}\right) \, . \end{aligned}$$

It should be noted here, that in particular, the finite element calculation program LS-Dyna^®always calculates the stresses during the simulation in a separate element coordinate system of the current configuration and also outputs them with respect to this coordinate system. Therefore, stresses and strains must first be transformed into the reference coordinate system and then into the tangential system in the current configuration. For a compact notation it is assumed in the following that all vectorial and tensorial quantities with the left superscript ${}^{u}$ and ${}^{\lambda}$ are already transformed to the tangential basis systems in the current state and adjoint configuration respectively.

We first consider solely the stress tensor ${}^{u}{\varvec{\sigma }}$ of the initial boundary value problem on the boundary of the hole and in the following only the adjoint linearized strain tensor ${}^{{{\lambda}}}{\varepsilon}$.

The reduced cross-sectional forces and moments per unit length can be determined with (numerical) integration from the stresses

$$\begin{aligned} {}^{u}{{f_{\alpha \beta }}}{}&= \int \limits _{-\frac{h}{2}}^{\frac{h}{2}} {}^{u}{{\sigma _{\alpha \nonscript \beta }}}{}\left( z\right) \,\text {d}z \, , \end{aligned}$$

(50)

$$\begin{aligned} {}^{u}{{m_{\alpha \beta }}}{}&= \int \limits _{-\frac{h}{2}}^{\frac{h}{2}} {}^{u}{{\sigma _{\alpha \nonscript \beta }}}{}\left( z\right) z \,\text {d}z \, , \end{aligned}$$

(51)

where the lower indices $\alpha$ and $\beta$ represent the combinations of the x and y components of the stress tensor.

The numerical integration with respect to the thickness coordinate z is done, for example, with the Lobatto integration.

We assume, that the stress can be split into a bending part and an in-plane stress part:

$$\begin{aligned} \sigma _{\alpha \beta }\left( z\right) = \sigma _{{\alpha \nonscript \beta }_{\left( P\right) }} \left( z\right) + \sigma _{{\alpha \beta }_{\left( S\right) }} \, . \end{aligned}$$

With the (in thickness direction) linear stress distribution (denoted by the lower index ${}_{(P)}$)

$$\begin{aligned} {}^{u}{{ \sigma _{\alpha \beta _{\left( P\right) }} }}{} = \frac{12\, {}^{u}{{m_{\alpha \nonscript \beta }}}{} }{h^3} z \end{aligned}$$

and the constant in-plane stresses (denoted by the lower index ${}_{(S)}$)

$$\begin{aligned} {}^{u}{{ \sigma _{\alpha \beta _{\left( S\right) }} }}{} = \frac{ {}^{u}{{f_{\alpha \beta }}}{} }{h} \, , \end{aligned}$$

(52)

the assumed stress distribution leads exactly to the reduced cross-sectional forces and moments in (50) and (51).

For the case, that the homogeneous stress state in the unperturbed plate does not exceed the limitations for the linear case, that is, the notch effect that comes with the insertion of the hole, does not go beyond the yield stress on the boundary of the hole, the integral forms on the hole boundary may be solved purely analytically. In the former works the in-plane stress state and the Kirchhoff plate bending are always considered separately with Airy’s stress function and the complex potentials approach respectively (Schumacher 1995; Sokołowski and Zochowski 2001; Bojczuk and Mróz 2008; Amstutz and Novotny 2011). These solutions may be superposed.

The main difficulty is now to predict the stresses and the adjoint gradient on the boundary of the fictious hole for the case of material nonlinearities.

Therefore, a phenomenological material interpolation was developed in Sect. 6.1, which is now used to calculate the stress tensor on the hole boundary. Using the interpolations of the effective Poisson’s ratio (36) and the tangent modulus (37), a discrete value is obtained for each thickness integration point depending on the effective plastic strain ${\varepsilon _{\text {eff}}^{pl}}$. For the “average elastic behavior”, the average tangent modulus and the average effective Poisson’s ratio over all thickness integration points $n_z$ are used

$$\begin{aligned} \bar{E}_{\text {T}}:=&\frac{1}{n_{z}} \sum \limits _{k=1}^{n_{z}} E_{\text {T}}\left( \frac{h}{2} {\xi _3}_{k} \right) \, , \nonumber \\ \bar{\nu }:=&\frac{1}{n_{z}} \sum \limits _{k=1}^{n_{z}} \nu _{\text {eff}}\left( \frac{h}{2} {\xi _3}_{k} \right) \, , \nonumber \\ \bar{\nu }^\prime :=&\frac{2\left( 1+ \bar{\nu }\right) }{3+\bar{\nu }}\, . \end{aligned}$$

(53)

The calculation of the circumferential stress for the in-plane stress part and for the bending part can be taken from the linear cases (Airy’s stress function and the complex potentials approach) and superposed with this material interpolation. Then, using the external stresses and moments, the stress at the hole boundary ${}^{u}{\sigma _{\varphi \nonscript \varphi }} \left( \varphi , z \right) \big |_{{\varGamma }_r}$ can be calculated by:

$$\begin{aligned} &{}^{u}{\sigma _{\varphi \nonscript \varphi }} \left( \varphi , z \right) \big |_{{\varGamma }_r} = {}^{u}{{ \sigma _{xx_{\left( S\right) }} }}{} \left( 1-2 \cos 2\varphi \right) \nonumber \\ & \quad + {}^{u}{{ \sigma _{yy_{\left( S\right) }} }}{} \left( 1+2 \cos 2\varphi \right) - 4 {}^{u}{{ \sigma _{xx_{\left( S\right) }} }}{} \sin 2 \varphi \nonumber \\& \quad + \frac{12}{h^3} z\left( {}^{u}{{m_{xx}}}{} \left( 1-\bar{\nu }^\prime \cos 2 \varphi \right) + {}^{u}{{m_{yy}}}{} \left( 1+\bar{\nu }^\prime \cos 2 \varphi \right) \right. \nonumber \\ &\left. \qquad + 2 {}^{u}{{m_{xy}}}{} \bar{\nu }^\prime \sin 2 \varphi \right) \,. \end{aligned}$$

(54)

The remaining components of the stress tensor at the boundary are zero, since the hole is free of external loads.

The cross-sectional forces and moments for the adjoint solution as suggested in the Fig. 7 on the right do not exist explicitly, because no stresses are calculated in the solution of the adjoint system. The adjoint gradient ${{\,\text{Grad}\,}}\varvec{{\lambda}}$ and the adjoint linearized strain tensor $\varvec{ {\varepsilon }}\left( \varvec{{\lambda}}\right)$ have been given mechanical names, but they are purely mathematical definitions without physical interpretation. We therefore now construct adjoint stresses to determine the adjoint linearized strain tensor on the hole boundary via the indirect route of the cross-sectional forces and moments and the stress function in the same manner as for the state solution.

We assume, that the same 5-parameter shell model holds for the adjoint in the shell. That means, the adjoint solution is kind of a midsurface-“displacement” $\varvec{ {\lambda }}$ and -“rotation” ${}^{\lambda}{\varvec{ {\theta }} }$ as already depicted with (49). This allows the calculation of the Rodrigues formula ${}^{\lambda }{\mathbf{{R}} }$ with (41), of the derivative of the rotation ${}^{\lambda }{{\mathbf{{R}}}_{,\alpha }}$ with (45) and finally of the director ${}^{\lambda}{\mathbf{{d}}}= {}^{\lambda }{\mathbf{{R}}} \mathbf{{N}}$ and of the derivative of the director ${}^{\lambda }{{\mathbf{{d}}}_{,\alpha }}$ with (44).

The definition of the covariant basis vectors (46) and (47) are also transferred to the current adjoint configuration:

$$\begin{aligned} {}^{\lambda }{\mathbf{{g}}_{\alpha }}&= {}^{\lambda }{\mathbf{{a}}_{\alpha }} + \frac{h}{2} \xi _3{}^{\lambda }{{\mathbf{{d}}}_{,\alpha }}\, ,\\ {}^{\lambda }{\mathbf{{g}}_{3}}&= \frac{h}{2} {}^{\lambda }{\mathbf{{d}}} \, . \end{aligned}$$

As the deformation gradient can also be calculated from the dyadic product of the co- and contravariant basis vectors of the shell (Wriggers 2008), one can determine the adjoint deformation gradient with

$$\begin{aligned} \varvec{F}\left( \varvec{ {\lambda }}\right) = {}^{\lambda }{\mathbf{{g}}_{i}}\otimes \mathbf{{G}}^{i}\, . \end{aligned}$$

With the definition of the displacement gradient (2) the adjoint gradient can finally be calculated with

$$\begin{aligned}{\text{Grad}\;{\varvec{\lambda}}}\left( z\right) = \varvec{F}\left( \varvec{ {\lambda }}\right) - \varvec{1}\, . \end{aligned}$$

This definition leads to the gradient in the global coordinate system and needs to be transformed to the current adjoint system

$$\begin{aligned} {}^{\lambda}{{{\,\text{Grad}\,}}\varvec{ {\lambda }}}\left( z\right) = {{}^{\lambda }{\mathbf{{E}}}{^{\text {loc}}}}^T {{\,\text{Grad}\,}}\varvec{ {\lambda }}\left( z\right) {{}^{\lambda }{\mathbf{{E}}}{^{\text {loc}}}}. \end{aligned}$$

The adjoint linearized strain tensor can then be calculated via the classical definition

$$\begin{aligned} {}^{\lambda }{{\varvec{ {\varepsilon }}}}{}\left( z\right) = \frac{1}{2}\left( {}^{\lambda }{{{\,\text{Grad}\,}}\varvec{ {\lambda }}}\left( z\right) + {}^{\lambda }{{{\,\text{Grad}\,}}\varvec{ {\lambda }}}\left( z\right) ^T \right) \, . \end{aligned}$$

Up to now, no assumptions considering the material behavior were made. Using the elasticity law for the Kirchhoff plate and the plane stress state, the in-plane adjoint stresses are calculated. The tangent modulus $E_{\text {T}}\left( z\right)$ from (37) based on the effective plastic strain of the initial boundary value problem is now used as the elastic modulus. The effective Poisson’s ratio $\nu _{\text {eff}}\left( z\right)$ is obtained from the step function (36). Both the tangent modulus and the effective Poisson’s ratio, depend on the thickness coordinate $z$. Thus, the adjoint stresses in Vogt’s notation are determined by

$$\begin{aligned} {}^{\lambda }{\varvec{ {\sigma }}} \left( z\right) = {}^{\lambda }{\mathbf{{D}}} \left( z\right) {}^{\lambda }{\varvec{ {\varepsilon }}}\left( z\right) \end{aligned}$$

(55)

with the adjoint elasticity matrix

$$\begin{aligned} &{}^{\lambda }{\mathbf{{D}}} \left( z\right) =\frac{E_{\text {T}}\left( z\right) }{1-\left( \nu _{\text {eff}}\left( z\right) \right) ^2} \begin{pmatrix} 1 &{} \nu _{\text {eff}}\left( z\right) &{} 0\\ \nu _{\text {eff}}\left( z\right) &{} 1 &{} 0\\ 0&{} 0&{} 1-\nu _{\text {eff}}\left( z\right) \\ \end{pmatrix}\,. \end{aligned}$$

Possibly occurring transverse shear stress will be neglected with the assumptions of the Kirchhoff plate theory. The stresses are the intermediate result to calculate the stresses on the hole boundary from the external load. Now the adjoint cross-sectional forces and moments shown in Fig. 7 can be calculated. This is done analogously to (50) and (51) by integrating the adjoint stresses:

$$\begin{aligned} {}^{\lambda }{{f_{xx}}}{}&= \int \limits _{-\frac{h}{2}}^{\frac{h}{2}} {}^{\lambda }{{\sigma _{xx}}}{}\left( z\right) \,\text {d}z \, , \end{aligned}$$

(56)

$$\begin{aligned} {}^{\lambda }{{m_{xx}}}{}&= \int \limits _{-\frac{h}{2}}^{\frac{h}{2}} {}^{\lambda }{{\sigma _{xx}}}{}\left( z\right)\, z \,\text {d}z\, . \end{aligned}$$

(57)

Also for the adjoint section forces the integration with respect to the thickness coordinate is done numerically for example with the Lobatto integration. From the forces, the average stresses for the in-plane stress state ${}^{\lambda }{{ \sigma _{{\alpha \beta }_{\left( S\right) }} }}{}$ are calculated as in (52). The adjoint strain in circumferential direction on the boundary of the hole ${}^{\lambda}{\varepsilon _{\varphi \nonscript \varphi }} \left( \varphi , z \right) \big |_{{\varGamma }_r}$ can then be calculated in the same way as the state stresses are calculated with the average tangent modulus $\bar{E}_{\text {T}}$ and the average effective Poisson’s ratio $\bar{\nu }$ and $\bar{\nu }^\prime$ from (53). With the inversion of (55) this yields

$$\begin{aligned}&{}^{\lambda}{\varepsilon _{\varphi \nonscript \varphi }} \left( \varphi , z \right) \big |_{{\varGamma }_r} = \nonumber \\&\frac{1}{\bar{E}_{\text {T}}} \Bigg[ {}^{\lambda }{{ \sigma _{{xx}_{\left( S\right) }} }}{} \left( 1-2 \cos 2\varphi \right) + {}^{\lambda }{{ \sigma _{{yy}_{\left( S\right) }} }}{} \left( 1+2 \cos 2\varphi \right) \nonumber \\&\qquad - 4 {}^{\lambda }{{ \sigma _{{xy}_{\left( S\right) }} }}{} \sin 2 \varphi \nonumber \\& \qquad + \frac{12}{h^3} z\left( {}^{\lambda }{{m_{xx}}}{} \left( 1-\bar{\nu }^\prime \cos 2 \varphi \right) \right. \nonumber \\& \qquad \left. + {}^{\lambda }{{m_{yy}}}{} \left( 1+\bar{\nu }^\prime \cos 2 \varphi \right) + 2 {}^{\lambda }{{m_{xy}}}{} \bar{\nu }^\prime \sin 2 \varphi \right) \Bigg] \, . \end{aligned}$$

(58)

The missing components of the adjoint strain tensor disappear in the scalar product with the vanishing radial state stresses and shear stresses on the boundary ${}^{u}{\sigma _{rr}} \left( \varphi , z \right) \big |_{{\varGamma }_r} = {}^{u}{\sigma _{r\varphi }} \left( \varphi , z \right) \big |_{{\varGamma }_r} =0$.

With the assumption (48), that the hole deformation can be neglected, the stress and adjoint strain approximations (54) and (58) can be merged and integrated with respect to the cylindrical parameterization of the hole boundary:

$$\begin{aligned}&- \left( \frac{\,\text {d}|c_r(\mathbf{{X}}_0)|}{\,\text {d}r}\right) ^{-1} \int \limits _0^T\int \limits _{{\varGamma }_r}\varvec{F}\varvec{S} \cdot {{\,\text{Grad}\,}}\varvec{ {\lambda }} \,\text {d}{\varGamma }\,\text {d}t\nonumber \\&\approx - \frac{1}{2 \pi rh}\int \limits _0^T\int \limits _{{\varGamma }_r}{}^{u}{\sigma _{\varphi \varphi }} {}^{\lambda}{\varepsilon }_{\varphi \varphi } \,\text {d}{\varGamma }\,\text {d}t\nonumber \\&= - \int \limits _0^T\Bigg [ \frac{1}{\bar{E}_{\text {T}}} \Bigg ( \left( {}^{u}{\sigma _{{xx}_{\left( S\right) }}}\! +\! {}^{u}{\sigma _{{yy}_{\left( S\right) }}} \right) \left( {}^{\lambda }{\sigma _{{xx}_{\left( S\right) }}}\! + \! {}^{\lambda }{\sigma _{{yy}_{\left( S\right) }}} \right) \nonumber \\&\qquad \Bigg . + 2 \left( {}^{u}{\sigma _{{xx}_{\left( S\right) }}} \! -\! {}^{u}{\sigma _{{yy}_{\left( S\right) }}} \right) \left( {}^{\lambda }{\sigma _{{xx}_{\left( S\right) }}} \!-\! {}^{\lambda}{\sigma _{{yy}_{\left( S\right) }}} \right) \nonumber \\&\qquad \Bigg . + 8 {}^{u}{\sigma _{{xy}_{\left( S\right) }}} {}^{\lambda}{\sigma _{{xy}_{\left( S\right) }}} \Bigg ) \nonumber \\&\qquad + \frac{6}{\bar{E}_{\text {T}}h^4} \Bigg( 2 \left( {}^{u}{m_{xx}} + {}^{u}{m_{yy}} \right) \left( {}^{\lambda }{m_{xx}} + {}^{\lambda }{m_{yy}}\right) \nonumber \\&\qquad \qquad \qquad + \frac{4\left( 1+ \bar{\nu }\right) ^2}{\left( 3+\bar{\nu }\right) ^2} \left( {}^{u}{m_{xx}} - {}^{u}{m_{yy}} \right) \left( {}^{\lambda}{m_{xx}} - {}^{\lambda}{m_{yy}} \right) \nonumber \\&\qquad \qquad \qquad \Bigg . + \left. \frac{16\left( 1+ \bar{\nu }\right) ^2}{\left( 3+\bar{\nu }\right) ^2} {}^{u}{m_{xy}} {}^{\lambda}{m_{xy}} \right) \Bigg ] \,\text {d}t\, . \end{aligned}$$

(59)

For each single point in the discretization of the time domain and for each integration point in the element midsurface the stress and the adjoint strain tensor is calculated for each thickness integration point. This is done for each element of the finite element discretization. Based on these tensors, the cutting forces and moment for the initial boundary value problem and the adjoint final value problem are determined. Finally, the integration in the time domain is done numerically for example with the trapezoidal rule for each integration point of the midsurface.

6.2.3 Calculation model of the explicit derivative of the internal energy

For the explicit derivative of the internal energy

$$\begin{aligned} - \lim _{r\downarrow 0} \left( \frac{\,\text {d}|c_r(\mathbf{{X}}_0)|}{\,\text {d}r}\right) ^{-1} \int \limits _0^T\int \limits _{{\varGamma }_r}\varvec{F}\varvec{S}\cdot {{\,\text{Grad}\,}}\dot{\mathbf{{u}}} \,\text {d}{\varGamma }\,\text {d}t\end{aligned}$$

a similar approximation of the boundary integration is used. Again, we assume, that the hole and its deformation is sufficiently small, that we can use the Cauchy stress tensor on the boundary of the hole in the initial configuration. As all values depend on the state solution of the initial boundary value problem, we omit the left superscript in this subsection.

No statement can be made about the symmetric spatial velocity gradient $\varvec{d}$ on the boundary of the undeformed hole even with the linear formulas, since this tensor is defined in the current configuration. For the calculation of the explicit derivative of the internal energy, we assume for simplification that the symmetric spatial velocity gradient and the strain rate can be treated approximately the same and do not differ, in particular, in the integration on the undeformed hole of the boundary:

$$\begin{aligned} \varvec{d}&= \frac{1}{2}\left( {{\,\text{grad}\,}}\dot{\mathbf{{u}}} + {{\,\text{grad}\,}}^T \dot{\mathbf{{u}}}\right) \\&\approx \frac{1}{2}\left( {{\,\text{Grad}\,}}\dot{\mathbf{{u}}}+ {{\,\text{Grad}\,}}^T \dot{\mathbf{{u}}} \right) = \dot{\varvec{ {\varepsilon }}} \end{aligned}$$

and

$$\begin{aligned} \int \limits _{{\varGamma }_r}\varvec{F}\varvec{S} \cdot {{\,\text{Grad}\,}}\dot{\mathbf{{u}}} \,\text {d}{\varGamma }= \int \limits _{{\varGamma }_r\left( t\right) } \varvec{\sigma } \cdot \varvec{d}\,\text {d}{\varGamma }\approx \int \limits _{{\varGamma }_r}\varvec{\sigma }\cdot \dot{\varvec{ {\varepsilon }}} \,\text {d}{\varGamma }\, . \end{aligned}$$

With this assumption, we can refer to the previous result (59) of the derivative of the mixed adjoint energy. Instead of the adjoint strain tensor ${}^{\lambda }{\varepsilon _{\varphi \varphi }}$, the state strain rate $\dot{\varepsilon }_{\varphi \varphi }$ is integrated. Since the derivative is linear with respect to time, the results from Airy’s stress function and the complex potentials approach can be used for the stress rate $\dot{\sigma }_{{\alpha \beta }_{\left( S\right) }}$ as well as for the time derivatives of the moments $\dot{m}_{\alpha \beta }$ on the hole boundary. Again, we use the material interpolation from Sect. 6.1 and can specify the integral on the hole boundary

$$\begin{aligned} \int \limits _{{\varGamma }_r}&\sigma _{\varphi \varphi } \dot{\varepsilon }_{\varphi \varphi } \,\text {d}{\varGamma }\nonumber \\&= \int \limits _{-\frac{h}{2}}^{\frac{h}{2}} \int \limits _0^{2\pi } \sigma _{\varphi \varphi } \dot{\varepsilon }_{\varphi \varphi } \, r\,\text {d}\varphi \,\text {d}z\nonumber \\&= \frac{1}{\bar{E}_{\text {T}}} \int \limits _{-\frac{h}{2}}^{\frac{h}{2}} \int \limits _0^{2\pi } \sigma _{\varphi \varphi } \dot{\sigma }_{\varphi \varphi } \, r\,\text {d}\varphi \,\text {d}z\nonumber \\&= 2 \pi rh\frac{1}{\bar{E}_{\text {T}}} \Big ( 3 \sigma _{{xx}_{\left( S\right) }} \dot{\sigma }_{{xx}_{\left( S\right) }} - \sigma _{{xx}_{\left( S\right) }}\dot{\sigma }_{{yy}_{\left( S\right) }} \Big . \nonumber \\&\Big . \qquad - \sigma _{{yy}_{\left( S\right) }}\dot{\sigma }_{{xx}_{\left( S\right) }} + 3 \sigma _{{yy}_{\left( S\right) }}\dot{\sigma }_{{yy}_{\left( S\right) }} + 8 \sigma _{{xy}_{\left( S\right) }}\dot{\sigma }_{{xy}_{\left( S\right) }} \Big ) \nonumber \\&\quad + 2 \pi rh\frac{6}{\bar{E}_{\text {T}}h^4}\Big ( 2 \left( m_{xx} + m_{yy} \right) \left( \dot{m}_{xx} + \dot{m}_{yy}\right) \Big . \nonumber \\&\qquad + \left( \bar{\nu }^\prime \right) ^2 \left( m_{xx} - m_{yy} \right) \left( \dot{m}_{xx} - \dot{m}_{yy}\right) \nonumber \\&\qquad + \Big . 4\left( \bar{\nu }^\prime \right) ^2 m_{xy} \dot{m}_{xy} \Big )\, . \end{aligned}$$

(60)

The evaluation of the above formula requires the strain rates from the solution of the boundary value problem, either directly from the finite element calculation of from the manual calculation from the strains with a detailed knowledge of the particular material model. If this is not the case (in LS-Dyna^® it is not possible to get the stress rates from the solver) one has the opportunity to approximate the stress rates via the difference quotient of successive stresses or to neglect the path dependency on the deformation history, which will be discussed in the following.

By means of an arbitrary component of the in-plane stresses from (60), the specific energy is the work that is performed by the stress $\sigma _{{\alpha \beta }_{\left( S\right) }}$, to get from one strain state ${\varepsilon _{{\alpha \beta }_{\left( S\right) }}\left( 0\right) }$ to the other strain state ${\varepsilon _{{\alpha \beta }_{\left( S\right) }}\left( T\right) }$. This is outlined by the area under the graph in Fig. 9.

Fig. 9

Neglection of the path dependency of the stresses on the deformation history

With substitution and partial integration in the time domain we get

$$\begin{aligned} &\int \limits _{\varepsilon _{{\alpha \beta }_{\left( S\right) }}\left( 0\right) }^{\varepsilon _{{\alpha \beta }_{\left( S\right) }}\left( T\right) } \sigma _{{\alpha \beta }_{\left( S\right) }} \,\text {d}\varepsilon _{{\alpha \beta }_{\left( S\right) }} = \int \limits _0^T\frac{1}{\bar{E}_{\text {T}}\left( t\right) } \sigma _{{\alpha \beta }_{\left( S\right) }} \dot{\sigma }_{{\alpha \beta }_{\left( S\right) }} \,\text {d}t\\ &= \frac{1}{2}\Bigg [ \underbrace{ \frac{1}{\bar{E}_{\text {T}}} \sigma ^2_{{\alpha \beta }_{\left( S\right) }}}_{\varepsilon _{{\alpha \beta }_{\left( S\right) }} \sigma _{{\alpha \beta }_{\left( S\right) }} }\Bigg ]_{t= 0}^{t=T} \! \underbrace{- \frac{1}{2}\int \limits _0^T\frac{\partial }{\partial t}\left( \frac{1}{\bar{E}_{\text {T}}\left( t\right) } \right) \sigma ^2_{{\alpha \beta }_{\left( S\right) }} \,\text {d}t}_{\text {neglected}} \,, \end{aligned}$$

where the remaining time integral will be neglected. That means, in Fig. 9, only the gray triangular area is kept and the hatched area, that defines the path dependency, is neglected. With this assumption, all terms in (60) are simplified. The explicit derivative of the internal energy can thereby be calculated by the cross-sectional forces and moments from the final state of the solution of the initial boundary value problem:

$$\begin{aligned} -&\lim _{r\downarrow 0} \left( \frac{\,\text {d}|c_r(\mathbf{{X}}_0)|}{\,\text {d}r}\right) ^{-1} \int \limits _0^T\int \limits _{{\varGamma }_r}{}^{u}{\varvec{F}} {}^{u}{\varvec{S}} \cdot {{\,\text{Grad}\,}}\dot{\mathbf{{u}}} \,\text {d}{\varGamma }\,\text {d}t\nonumber \\&\approx - \frac{1}{2 \pi rh} \int \limits _0^T\int \limits _{{\varGamma }_r}\sigma _{\varphi \varphi } \dot{\varepsilon }_{\varphi \varphi } \,\text {d}{\varGamma }\,\text {d}t\nonumber \\&= - \frac{1}{2\bar{E}_{\text {T}}\left( T\right) } \Bigg ( \left( \sigma _{{xx}_{\left( S\right) }}\left( T\right) + \sigma _{{yy}_{\left( S\right) }}\left( T\right) \right) ^2 \nonumber \\&\qquad \qquad \quad + 2 \left( \sigma _{{xx}_{\left( S\right) }}\left( T\right) - \sigma _{{yy}_{\left( S\right) }}\left( T\right) \right) ^2 \Bigg . \nonumber \\&\qquad \qquad \quad \Bigg . + 8 \sigma _{{xy}_{\left( S\right) }}^2\left( T\right) \Bigg ) \nonumber \\&\quad - \frac{3}{\bar{E}_{\text {T}}\left( T\right) h^4} \Bigg (2 \big ( m_{xx}\left( T\right) + m_{yy}\left( T\right) \big )^2 \Bigg . \nonumber \\&\qquad \qquad \qquad + \frac{4\left( 1+ \bar{\nu }\right) ^2}{\left( 3+\bar{\nu }\right) ^2} \big ( m_{xx}\left( T\right) - m_{yy}\left( T\right) \big )^2 \nonumber \\&\qquad \qquad \qquad \Bigg . + \frac{16\left( 1+ \bar{\nu }\right) ^2}{\left( 3+\bar{\nu }\right) ^2} m_{xy}^2\left( T\right) \Bigg )\, . \end{aligned}$$

(61)

Just like the formula for the explicit derivative of the mixed adjoint energy in (59), both summands in (61) are multiplied by the reciprocal tangent modulus $1/\bar{E}_{\text {T}}(t)$. This tangent modulus takes values from the elastic young’s modulus $E$ to almost zero at full plastic strain in all thickness integration points, cf. Fig. 3c. With increasing plastic strain, the reciprocal tangent modulus becomes larger and the resulting topological derivative becomes larger than with purely elastic behavior. This effect has been shown in Weider and Schumacher (2018) with numerical simulations of a plate with a hole under biaxial tension and with an ideal plastic tangent modulus of zero.

This calculation model has the largest factor of uncertainty and potential for improvement at once. By use of the true stress rates the path dependency could be kept and for the case of unloading at the final time this would be the only way of a reliable calculation.

6.2.4 Calculation model of the explicit derivatives of the inertia, dam** and volume forces

The remaining material derivatives in (35) include directly the state solution of the initial value boundary problem $\mathbf{{u}}$ and its derivatives with respect to the time $\dot{\mathbf{{u}}}$ and $\ddot{\mathbf{{u}}}$ and the adjoint solution $\varvec{ {\lambda }}$. We present here the calculation model for the inertia term

$$\begin{aligned} - \lim _{r\downarrow 0} \left( \frac{\,\text {d}|c_r(\mathbf{{X}}_0)|}{\,\text {d}r}\right) ^{-1} \int \limits _0^T\int \limits _{{\varGamma }_r}\varrho \ddot{\mathbf{{u}}} \cdot \varvec{ {\lambda }} \,\text {d}{\varGamma }\,\text {d}t\end{aligned}$$

(62)

in detail. The further derivatives can be simply derived from that result.

The density $\varrho$ is constant in the whole body and therefore serves as a scaling factor. The state acceleration consists of two components, that are illustrated in Fig. 10. Figure 10a shows the global kinematic behavior, that is driven by the behavior of the microcell. The whole microcell and therefore also the cut-out boundary is undergoing the translational and rotational state acceleration and adjoint of the point $\mathbf{{X}}_0$. Caused by the state acceleration and the adjoint, the boundary of the hole is locally deformed, as it is shown exemplary in Fig. 10b with the adjoint of the midsurface.

Fig. 10

Visualization of the derivative of the inertia term

This distortion is from the same difficulty as the integral on the deformed boundary in the former sections. We do not exactly know the adjoint displacement (and the state acceleration) on the boundary. From the strain interpolations in the Sects. 6.2.2 and 6.2.3 it could in principle be possible to get the displacement by the formula of Cesáro (1906) with a huge numerical effort. We assume here for simplicity, that the inertia effect of the hole deformation is sufficiently small compared to the inertia behavior caused by the microcell translation and rotation and can be neglected. The state acceleration and the adjoint on the boundary of the hole are reduced to the global microcell behavior:

$$\begin{aligned} \left. \ddot{\mathbf{{u}}} \right| _{{\varGamma }_r} = \ddot{\mathbf{{u}}} \left( \mathbf{{X}}_0 \right) \qquad \text {and} \qquad \left. \varvec{ {\lambda }} \right| _{{\varGamma }_r} = \varvec{ {\lambda }} \left( \mathbf{{X}}_0 \right) \, . \end{aligned}$$

The translational components are the state acceleration of the midsurface ${}^{u}{\ddot{\mathbf{{x}}}}_\mathcal {M}$ and the adjoint of the midsurface ${}^{\lambda }{\mathbf{{x}}}_\mathcal {M}$. The rotational components are the state acceleration of the director ${}^{u}{\ddot{\mathbf{{d}}}}$ and the adjoint director ${}^{\lambda }{\mathbf{{d}}}$. For the calculation of the state acceleration of the director we first calculate the state velocity of the director

$$ \dot{\mathbf{d}} = \dot{\mathbf {R}} \mathbf{N}+ \mathbf{R}\dot{\mathbf{N}}=\dot{\mathbf {R}} \mathbf{N}. $$

As the Rodrigues formula is in fact a matrix exponential of the cross-product matrix $[\,\varvec{ {\theta }}\,]_{\textsf {x}}$, its derivative with respect to time is given by

$$\begin{aligned} \dot{\mathbf{ {{R}}}} = [\dot{\varvec{ {\theta }}}\left( t\right) ]_{\textsf {x}} \mathbf{{R}}\left( \varvec{ {\theta }}\left( t\right) \right) \, . \end{aligned}$$

(63)

this yields $\ddot{\mathbf{{d}}} = \ddot{\mathbf{ {{R}}}} \mathbf{{N}}$ for the second derivative with respect to time with

$$\begin{aligned} \ddot{\mathbf{ {{R}}}}&= [\,\ddot{\varvec{ {\theta }}}\,]_{\textsf {x}} \mathbf{{R}}\left( \varvec{ {\theta }}\right) + [\,\dot{\varvec{ {\theta }}}\,]_{\textsf {x}} \dot{\mathbf{ {{R}}}}\left( \varvec{ {\theta }}\right) \nonumber \\&= \left( [\,\ddot{\varvec{ {\theta }}}\,]_{\textsf {x}} + [\,\dot{\varvec{ {\theta }}}\,]_{\textsf {x}} [\,\dot{\varvec{ {\theta }}}\,]_{\textsf {x}} \right) \mathbf{{R}}\left( \varvec{ {\theta }}\right) \, . \end{aligned}$$

(64)

The rotational velocity $\dot{\varvec{ {\theta }}}$ and the rotational acceleration $\ddot{\varvec{ {\theta }}}$ are calculated within the solution of the initial boundary value problem by the Newmark-approximations.

The displacement in the shell $\mathbf{{u}}$ is given by the definition of the displacement (40) together with the special shell formulation of the initial and the current configuration (38) and (39)

$$\begin{aligned} \mathbf{{u}} \left( \xi _{\alpha }, \xi _3, t\right) = {}^{u}{\mathbf{{x}}_\mathcal {M}} \left( \xi _{\alpha }, t\right) - \varvec{ {\mathbf{{X}}}}_\mathcal {M}\left( \xi _{\alpha }\right)+ \frac{h}{2} \xi _3\left( {}^{u}{\mathbf{{d}}}\left( \xi _{\alpha }, t\right) - \mathbf{{N}}\left( \xi _{\alpha }\right) \right) \,. \end{aligned}$$

As $\mathbf{{X}}_\mathcal {M}$ and $\mathbf{{N}}$ are constant in time, the double time derivative with the acceleration of the Rodrigues formula $\ddot{\varvec{ {\mathbf{R}}}}$ from (64) leads to

$$\begin{aligned} \ddot{\mathbf{{u}}} = {}^{u}{\ddot{\mathbf{{x}}}}_\mathcal {M}+ \frac{h}{2} \xi _3{}^{u}{\ddot{\mathbf{{d}}}} = {}^{u}{\ddot{\mathbf{{x}}}}_\mathcal {M}+ \frac{h}{2} \xi _3{}^{u}{\ddot{\varvec{ {\mathbf{R}}}}}\mathbf{{N}}\, . \end{aligned}$$

(65)

For the adjoint, the shell formulation is used likewise:

$$\begin{aligned} \varvec{ {\lambda }} \left( \xi _{\alpha }, \xi _3, t\right) =& {}^{\lambda }{\mathbf{{x}}}_\mathcal {M}\left( \xi _{\alpha }, t\right) - \varvec{ {\mathbf{{X}}}}_\mathcal {M}\left( \xi _{\alpha }\right) \nonumber \\ &+ \frac{h}{2} \xi _3\left( {}^{\lambda}{\mathbf{{d}}}\left( \xi _{\alpha }, t\right) - \mathbf{{N}}\left( \xi _{\alpha }\right) \right) \,. \end{aligned}$$

(66)

The calculation of the adjoint director ${}^{\lambda }{\mathbf{{d}}} = {}^{\lambda }{\mathbf{{R}}}\mathbf{{N}}$ was already described in Sect. 6.2.2, where the covariant basis vectors in the current adjoint configuration were calculated.

Inserting the acceleration (65) and the adjoint (66) into the inertia integral (62), results in constant and quadratic terms with respect to the shell thickness $h$. All terms are constant in radius $r$ and angle $\varphi$ of the polar transformation of the hole boundary and therefore the $2\pi r$ vanishes with the derivative of the cylinder volume (12). Hence, a translational and rotational part remain for the calculation of the inertia term:

$$\begin{aligned} &- \lim _{r\downarrow 0} \left( \frac{\,\text {d}|c_r(\mathbf{{X}}_0)|}{\,\text {d}r}\right) ^{-1} \int \limits _0^T\int \limits _{{\varGamma }_r}\varrho \ddot{\mathbf{{u}}} \cdot \varvec{ {\lambda }} \,\text {d}{\varGamma }\,\text {d}t\nonumber \\& = - \int \limits _0^T\Bigg ( \varrho \ddot{\mathbf{{u}}}_\mathcal {M} \cdot \varvec{ {\lambda }}_\mathcal {M} \Bigg .\nonumber -\Bigg . \frac{\varrho h^2}{12} \left( {}^{u}{\ddot{\varvec{ {\mathbf{R}}}}}\mathbf{{N}} \cdot \mathbf{{N}} - {}^{u}{\ddot{\varvec{ {\mathbf{R}}}}} \mathbf{{N}} \cdot {}^{\lambda}{\mathbf{{d}}} \right) \Bigg ) \,\text {d}t\,. \end{aligned}$$

(67)

This result can be interpreted as the specific kinetic translational and rotational energy of the cylindrical cut-out. The energy is thereby induced by the adjoint, as it is also the case for the adjoint energy. The factor $\frac{\varrho h^2}{12}$ corresponds to the moment of inertia of a thin rod—to which the cut-out is reduced with the transition to the limit—related to the volume that rotates about its transverse axis.

With the similar approach and the translational midsurface velocity $\dot{\mathbf{{u}}}_\mathcal {M}$ and the rotational velocity ${}^{u}{\dot{\varvec{ {\mathbf{R}}}}}$ from (63) we get for the derivative of the dam** term

$$\begin{aligned} & \lim _{r\downarrow 0} \left( \frac{\,\text {d}|c_r(\mathbf{{X}}_0)|}{\,\text {d}r}\right) ^{-1} \int \limits _0^T\int \limits _{{\varGamma }_r}c_d\dot{\mathbf{{u}}} \cdot \varvec{ {\lambda }} \,\text {d}{\varGamma }\,\text {d}t\nonumber \\ & = \int \limits _0^T\Bigg (c_d\dot{\mathbf{{u}}}_\mathcal {M} \cdot \varvec{ {\lambda }}_\mathcal {M} \Bigg . \nonumber \Bigg . - \frac{c_dh^2}{12} \left( {}^{u}{\dot{\mathbf{ {{R}}}}}\mathbf{{N}} \cdot \mathbf{{N}} - {}^{u}{\dot{\mathbf{ {{R}}}}} \mathbf{{N}} \cdot {}^{\lambda }{\mathbf{{d}}} \right) \Bigg ) \,\text {d}t\,. \end{aligned}$$

(68)

As the volume forces are usually only translational forces, the explicit derivative of the volume forces leads to a dependency on the midsurface adjoint:

$$\begin{aligned} \lim _{r\downarrow 0} \left( \frac{\,\text {d}|c_r(\mathbf{{X}}_0)|}{\,\text {d}r}\right) ^{-1} \int \limits _0^T\int \limits _{{\varGamma }_r}\mathbf{{f}} \cdot \varvec{ {\lambda }} \,\text {d}{\varGamma }\,\text {d}t\nonumber \\ \quad= \int \limits _0^T\mathbf{{f}} \cdot \varvec{ {\lambda }}_\mathcal {M} \,\text {d}t\,. \end{aligned}$$

(69)

6.2.5 Summary of the computational procedure

The analytical descriptions of stress and strain on the hole boundary assuming linear material behavior and small displacements quickly reach the limit of validity because of the plastic strain initiated by the notch effect. With the material interpolation in Sect. 6.1, a phenomenological modeling for the temporary linear material behavior was defined, which takes into account the local plastic strains and the associated artificial change of the material. Based on the internal forces of the microcell, the calculation models of the derivative of the mixed adjoint energy and the explicit derivative of the internal energy were explained using the linearized formulas. Finally, the calculation of the derivative of the inertia term is based on the mean surface translation and rotation. The calculation models of this section are summarized in Algorithm 3.

The calculation of the topological derivative is performed in three major steps in a row. The solution of the initial boundary value problem with implicit time integration is followed by the computation of the adjoint with one of the adjoint solution schemes, either “first differentiate then discretize” from Algorithm 1 or “first discretize then differentiate” from Algorithm 2. The solution for the state $\mathbf{{u}}$ and the adjoint $\varvec{ {\lambda }}$ is made simultaneously for all degrees of freedom with the finite element method, whereas the topological derivative is subsequently calculated pointwise with Algorithm 3. This sequence is illustrated in Fig. 11.

Algorithm 3 Numerical evaluation of the boundary integrals and calculation of the topological derivative

Fig. 11

Procedure for the calculation of the topological derivative

In this paper, the numerical evaluation of the boundary integrals is focused in particular on shells. The material interpolation as well as the principal ideas and assumptions can similarly be transferred to volumetric modelling with full dimensional spherical cut-outs. The stresses and strains on the boundary of the hole can then be calculated via the solution of Léon’s problem (Leon 1908). The section forces are determined by 3 normal and 3 shear stresses and no moments are needed.

6.3 Discussion of simplifying assumptions

The presented computational concept is based on simplifying assumptions. Motivated by the procedure of implicit time integration for the solution of the initial boundary value problem and the solution of the adjoint final value problem with the tangential stiffness matrices, a temporary linear material behavior is assumed. The material interpolation was developed to calculate the cross-sectional forces and moments. And likewise, the stresses and strains on the boundary of the hole were following from the section forces with the linear formulas of Airy’s stress function and the complex potentials approach. This phenomenological material interpolation can be improved based on the numerical material modeling. A closed formulation for the stresses and strains on the boundary of the hole in the case of nonlinear material behavior offers the greatest potential for the correct and efficient calculation for the nonlinear topological derivative. Here, the investigations of Sawin (1956) with the solutions for ideal plastic material behavior without hardening do not yet go far enough.

In the calculation of the stresses and strains on the boundary of the hole, the formulas based on Kirchhoff’s plate theory were used. This neglects the transverse shear deformation, which a shell formulation in nonlinear continuum mechanics, however, has in principle. In the case of very small holes in relation to the shell thickness (in the case of the topological derivative with infinitesimal hole radius), the Kirchhoff assumption is no longer sufficient for the stress estimation on the hole boundary. Further analytical developments with the Reissner-Mindlin plate theory by Felger and Becker (2019) would improve the calculation model to more accuracy.

In the derivation of the mixed adjoint energy as well as in the calculation model of the explicit derivative of the internal energy, the formulation in the current configuration was initially considered in order to fall back on quantities with the Cauchy stress tensor and the linearized strain tensor, which can be calculated from the linear formulas for the circumferential stress on the hole boundary. However, the integration was finally not performed on the hole boundary in the current configuration but in the initial configuration, since the parameterization of the cylinder shell surface is particularly simple here. The lack of knowledge of the exact deformation of the hole also leads to the loss of the inertia change due to the change of the hole shape when calculating the derivative of the inertia term by neglecting the kinematic behavior of the hole boundary. In both cases, using the formula of Cesáro (1906) by integrating the strain, the change of the cut-out boundary could be calculated, but again this would be based on geometrical linearity.

In the calculation model of the explicit derivative of the internal energy, an independence of the internal energy from the deformation history was assumed for simplification. Here, the stress rates were missing for improved evaluation in particular when using LS-Dyna^® for the solution of the initial boundary value problem. This can be improved with an output of the finite element calculation program or at least a suitable numerical differentiation from the time histories of the stresses.

A more general solution, which allows an evaluation of the boundary integrals independent of the linear circumferential stresses, is in general desirable. For the explicit derivation of the internal energy, it was proposed in Weider and Schumacher (2016) to create an approximation for the hole boundary integral as a function of the section forces on the basis of the separate calculation of the microcell. For the derivation of the mixed adjoint energy with the stress and the adjoint gradient, there are more quantities on the hole boundary, which have to be predicted by the approximation as a function of all cross-sectional forces and moments. A modeling with artificial neural networks would be a possible solution, since with a neural network several quantities can be approximated at once.

7 Examples

We consider a 1-mm-thick cantilever of length 150 mm and width 100 mm clamped on one side and loaded with a series of individual forces on a width of 25 mm (cf. Fig. 12). Using two loadcases and with two positions of the displacement functional each, the topological derivative for the displacement of a single point is discussed. In the first loadcase, the force is applied to the nodes at $X_1=150$ mm, in the second loadcase to the nodes at $X_1 = 100$ mm. On the outer nodes of the load application line, the force is scaled by 0.5, so that a total force of $F=0.4$ kN is applied linearly over the loading period 25 ms. The single point displacement is measured at the two marked red points $\mathbf{{X}}_*=\left( 75.0, 0.0, 0.0\right) ^T$ and $\mathbf{{X}}_* = \left( 150.0, -\,37.5, 0.0\right) ^T$ in negative $\mathbf{{e}}_3$-direction ($\mathbf{{v}} = \left( 0.0, 0.0, -1.0\right) ^T$).

The beam is discretized in space with 1536 elements with an element edge length of 3.125 mm. The numerical integration is carried out at 4 integration points in the shell midsurface (ELFORM 16) and 9 thickness integration points (NIP 9) with the integration rule of Lobatto. All degrees of freedom are locked at the clamped edge. The implicit time integration is performed with a constant step size of $\varDelta t= 0.08712$ ms and the Newmark constants $\beta =0.5$ and $\gamma =0.25$.

Fig. 12

Geometry and loading conditions of the cantilever

7.1 Analysis of the 1st loadcase

The displacement history for the first loadcase and the single point displacement for $\mathbf{{X}}_\star =\left( 150.0, -37.5, 0.0\right) ^T$ is shown in Fig. 13 in timesteps of 5 ms. The maximum von Mises equivalent stress in the shell is documented with the color scheme. The yield stress for the material used is 0.173 GPa. Since no unloading of the cantilever is allowed here, the red area in the von Mises equivalent stress also corresponds to the area in which plastic strain occurs.

For the displacement of the red marked point in negative $\mathbf{{e}}_3$ direction, the pseudo loads $-\frac{\partial j_u}{\partial {\mathbf{{u}}^{(k)}}}$ are determined with (26) for $p=5$. Since these already include the integration weight ${w^{(k)}}$, the pseudo load vector at the final time point with the trapezoidal rule and constant time step size is only half as large as at the final time. The pseudo load vectors are plotted for selected time points as blue arrows in Fig. 13. The pseudo load vector is unitless and always acts at the point whose displacement is to be captured by the functional. In the figure, these vector arrows are plotted on the undeformed geometry to match the diagram. From the representation of the example, it is easy to see that the external load of the initial boundary value problem is irrelevant for the calculation of the pseudo loads, but only the displacement state is needed.

Fig. 13

True to scale deformation and loading of the cantilever in the 1st loadcase, single point displacement and pseudo load

7.1.1 Sensitivities by “first differentiate then discretize”

We first consider the sensitivities according to the adjoint solution scheme “first differentiate then discretize”. In Fig. 14 the graphs of the individual terms of the topological derivative of the internal energy are shown and colored according to the values. Because of the differences in magnitude, they are each shown with their own scales. The derivative of the mixed adjoint energy dominates the sensitivity, and the derivative of the inertia term is of small order. The free edges at $X_2 = \pm 50$ mm have only a low sensitivity. The plotted black line indicates the zero-value-contour line of the individual terms. Here it can be observed that this contour line does not extend to the force introduction.

This can be interpreted as follows: The topological derivative of the internal energy shows the load-bearing structure of the cantilever from the application of the force in two beams to the clam**. If we want to increase the energy absorption of the cantilever by inserting a hole, then it should be inserted in the red area. This would be tantamount to weakening the load-bearing capacity, there will be more strain as a result. If we do not want to weaken the beam, then we should insert a hole in the blue area, since this contributes least to the structural integrity for this load.

Fig. 14

Topological Derivative of the internal energy according to “first differentiate then discretize”, 1st loadcase (in $\text {kg}\cdot \text {mm}^{-1}\text {ms}^{-2}$)

Figure 15 shows the graph of the sensitivity terms for the single point displacement at ${\mathbf{{X}}_*=(75.0, 0.0, 0.0)^T}$ in negative $\mathbf{{e}}_3$-direction. The displacement of the point is supported by the same supporting structure as the internal energy. The single point displacement is intended to evaluate the intrusion into regions worth protecting. For example, to protect adjacent areas in the negative $\mathbf{{e}}_3$ direction, the insertion of a hole only in the area of the edges is reasonable. Minimizing single point displacement and maximizing internal energy are conflicting goals.

Fig. 15

Topological derivative of the single point displacement at $\mathbf{{X}}_*=\left( 75.0, 0.0, 0.0\right) ^T$ according to “first differentiate then discretize”, 1st loadcase (in $\text {mm}^{-2}$)

For the second displacement point at $\mathbf{{X}}_* = \left( 150.0, -\,37.5, 0.0\right) ^T$, the pseudo load was shown in Fig. 13. In the derivative of the inertia term of the topological derivative in Fig. 16, the imbalance due to the lateral pseudo load application is barely visible. More noticeable here is the high sensitivity of the bending region directly at the clamped edge. In the derivative of the mixed adjoint energy, there is a significant shift in the topological derivative and the restraint is “connected” to the location of the single point displacement.

Fig. 16

Topological derivative of the single point displacement at $\mathbf{{X}}_* = \left( 150.0, -37.5, 0.0\right) ^T$ according to “first differentiate then discretize”, 1st loadcase (in $\text {mm}^{-2}$)

7.1.2 Sensitivity by “first discretize then differentiate”

For comparison with the adjoint solution scheme “first discretize then differentiate”, the graphs of the derivative of the mixed adjoint energy, the derivative of the inertia and the graph of the topological derivative of the internal energy are shown in Fig. 17. The explicit derivative is the same for both schemes.

For a direct quantitative comparison, the difference of the two schemes is colored in Fig. 18. The zero crossings of the derivative of the mixed adjoint energy and the topological derivative are shown as black lines. The deviation in the derivative of the inertia term can also be seen in the Figs. 14 and 17. The divergence in the derivative of the mixed adjoint energy and the topological derivative is seen in the extrema. Along the zero crossing, the difference is zero, the maximum sensitivity after “first differentiate then discretize” is smaller than the maximum sensitivity “first discretize then differentiate”, for the minimum sensitivity it is the other way round.

Fig. 17

Topological derivative of the internal energy according to “first discretize then differentiate”, 1st loadcase (in $\text {kg}\cdot \text {mm}^{-1}\text {ms}^{-2}$)

Fig. 18

Difference of the topological derivative of the internal energy caused by the different adjoint schemes

7.2 Analysis and sensitivities of the 2nd loadcase

By shifting the load to the position at two-thirds of the cantilever’s length, the lever arm of the load is shortened and therefore, the internal energy is lower than in the 1st loadcase. The true to scale deformation and the loading are shown in Fig. 19. Plastic strain occurs practically only close to the mounting area. In the area between the applied load and the free edge, one still discovers a loading in the form of internal stresses, which is depicted in the lighter blue “shadow”.

Fig. 19

True to scale deformation and loading of the cantilever in the 2nd loadcase

The topological derivative of the internal energy according to the adjoint solution scheme “first differentiate then discretize” in Fig. 20 shows that the load-bearing structure within the beam also extends around the force application. Since the computational model presented in Sect. 6.2 is based on superposition of the plane stress state and plate bending as for the linear case, we can consider the contributions to the topological derivative separately. The topological derivative is dominated by the plane stress part (Fig. 21). The plate part, that arises from pure bending, is only about $\frac{2}{5}$ as large as the plane stress part and actually acts only in the mounting area. The plane stress part dominates the center of the cantilever and points in two rays from the load application to the mounting.

Fig. 20

Topological derivative of the internal energy according to “first differentiate then discretize”, 2nd loadcase (in $\text {kg}\cdot \text {mm}^{-1}\text {ms}^{-2}$)

Fig. 21

Bending plate and plane stress contribution to the topological derivative according to “first differentiate then discretize”, 2nd loadcase (in $\text {kg}\cdot \text {mm}^{-1}\text {ms}^{-2}$)

The sensitivity of the displacement functional of the point $\mathbf{{X}}_*=\left( 75.0, 0.0, 0.0\right) ^T$ in Fig. 22 shows a similar shape as the topological derivative of the internal energy. Again, the region of the cantilever beam whose sensitivity is greater than zero extends around the force application. The sides and the free end are unsurprisingly insignificant.

For a targeted softening of the cantilever, the topological derivative reveals the most sensitive region in the center of the beam directly before the force application. This region again contributes to the support of the single point at the center of the beam at ${\mathbf{{X}}_*=(75.0, 0.0, 0.0)^T}$, benefiting even more from the bending stiffness at the mounting. As expected, the side flanks outside the supporting triangle and the free end of the beam have little effect on the local displacement at the center.

Fig. 22

Topological derivative of the single point displacement at $\mathbf{{X}}_*=\left( 75.0, 0.0, 0.0\right) ^T$ according to “first differentiate then discretize”, 2nd loadcase (in $\text {mm}^{-2}$)

The topological derivative of the displacement functional of the point at the leading edge at $\mathbf{{X}}_* = \left( 150.0, -\,37.5, 0.0\right) ^T$ in Fig. 23 shows a different characteristic. The point is supported by a circumferential frame of the derivative of the mixed adjoint energy. Between mounting and force application, directly before the force application, the cantilever beam has the lowest sensitivity. Inserting a hole in the beam center does not affect the single point in the front at all. Or in other words: If the dark blue area here would be very soft, then the force would sink there and cause locally large strains, but the single point would be carried by the frame.

Fig. 23

Topological derivative of the single point displacement at $\mathbf{{X}}_* = \left( 150.0, -37.5, 0.0\right) ^T$ according to “first differentiate then discretize”, 2nd loadcase (in $\text {mm}^{-2}$)

The illustrations in Fig. 24 show the topological derivatives of the displacement functional for exponents $p=2$, $p=3$ and $p=4$ according to the adjoint solution scheme “first discretize then differentiate”. The support frame for the single point displacement is tightened with the exponent p. When the exponent is smaller, a longer time range of the load contributes to the determination of the sensitivity. As the exponent becomes larger, the influence of the maximum displacement becomes larger and the time range of the deformation history that influences the sensitivity of the single point becomes smaller. In the case of the cantilever, for a large exponent, this is the time range immediately before the end of the simulation.

Fig. 24

Topological derivative of the single point displacement at $\mathbf{{X}}_* = \left( 150.0, -37.5, 0.0\right) ^T$ according to “first discretize then differentiate”, 2nd loadcase (in $\text {mm}^{-2}$)

However, these interpretations are only a snapshot. For component design, the insertion of a hole must be done successively, for example in an optimization process. In conjunction with the level set algorithm of Amstutz and Andrä (2006) and the optimization scheme of Weider and Schumacher (2018), the Topological Derivative is promising for the topological optimization of crashloaded structures. In the aforementioned level set algorithm, the Topological Derivative can be used directly for the adjustment of the level set function. In the crash simulation only the upper level set has to be calculated and the derived sensitivities can be projected to the entire design space.

8 Conclusion

The topological derivative of a point is the sensitivity that describes how a functional changes when an infinitesimal cut-out is introduced into the mechanical body at that point. In this paper we derived a calculation scheme for the topological derivative considering large deformations and material nonlinearities with the adjoint method and the material derivative.

Two different solution schemes for the adjoint arise, depending on the order of time discretization and differentiation. In both cases, a boundary value problem with final value constraints arises, which is solved backwards from the final value and is linear in the mass matrix and the tangential stiffness matrix of the system at each discrete time point, even if the material behavior in the crash itself is nonlinear.

A numerical evaluation scheme using a temporary material linearization was shown for shell structures. With the application of the calculation scheme on the cantilever beam under highly dynamic loading, the plausibility for the semi-analytical sensitivities of the displacement functional was validated by varying the location of the load. If the location of the single point displacement to be measured is close to the force application, the sensitivity is significantly influenced by the general load path. If the distances between the force application and the single point are larger, the sensitivity may reveal altered support structures.

The basic research for topology optimization of crashloaded structures has thus been extended by a significant part. The semi-analytical sensitivities can be incorporated into optimization systems. In order to use the topological derivation in crash optimization, contact - i.e. the variability of the external forces due to displacement - must be taken into account. Here one can benefit from the latest publications of the density method. Then the application with the level set method for example is attractive for practical use.

As a suggestion for further work, the simplifying assumptions should be refined. The material interpolation can be improved by the knowledge of the material formulation during loading. Therefore, a better connection to the finite element calculation programs would be desirable. With knowledge of the stress rate at each integration point, the assumption of independence of the final stress from the deformation history can also be replaced by a time integral when calculating the explicit derivative of the internal energy.

More fundamental are the assumptions for the integration with respect to the cut-out boundary and its shape as well as the stress calculation on the cut-out boundary, which is based on the temporary linearized material behavior in the solution of the initial boundary value problem with implicit time integration. The numerical calculation of the boundary integrals is based on the linear elastic theory. An analytical solution for the stress prediction on the hole boundary under large displacement and nonlinear material behavior would complete the calculation of the topological derivative analytically.