# Stress and strain control via level set topology optimization

## Abstract

This paper presents a level set topology optimization method for manipulation of stress and strain integral functions in a prescribed region (herein called sub-structure) of a linear elastic domain. The method is able to deviate or concentrate the flux of stress in the sub-structure by optimizing the shape and topologies of the boundaries outside of that region. A general integral objective function is proposed and its shape sensitivities are derived. For stress isolation or maximization, a von Mises stress integral is used and results show that stresses in the sub-structure can be drastically reduced. For strain control, a strain integral combined with a vector able to select the component of the strain is used. A combination of both can be used to minimize deformation of a prescribed direction. Numerical results show that strain can be efficiently minimized or maximized for a wide range of directions. The proposed methodology can be applied to stress isolation of highly sensitive non strain-based sensors, design for failure, maximization of mechanical strain and strain direction control for strain-based sensors and microdevices.

## Keywords

Topology optimization Level-set method Stress isolation Strain control## 1 Introduction

Strains and stresses are mechanical quantities intrinsically present in a load carrying structure. Although directly related, strains and stresses can have different functionalities. In a statically loaded structure, low stress levels are generally preferred. On the other hand, higher stress or strain can be beneficial in certain microdevices or energy harvesters and in design for failure.

The optimization of stress-based problems has long been a challenge for shape and topology optimization (Duysinx and Bendsøe 1998; Duysinx et al. 2008; Le et al. 2010). To the best of our knowledge, Li and Wang (2014) were the first authors to conduct stress isolation via shape and topology optimization by applying a level set method. In their work, different stress limits are imposed in the base and sub-structure, with the stress allowance in the sub-structure much smaller than the base one. Recently, Luo et al. (2017) used a similar idea but a density-based method and nonlinear finite elements to design stress-isolated hyperelastic composite structures. Both of these works, however, do not account for strain.

Strain control can be useful for a variety of applications. For example, piezoelectric vibration energy harvesters can obtain higher electrical charges by maximizing their strain (Lin et al. 2011; Kiyono et al. 2016; Thein and Liu 2017). One way of maximizing the strain in a piezoelectric component is to optimize the layout of the component itself (Silva et al. 1997; Silva 2003; Xia et al. 2013). The other approach can be the maximization of the mechanical strain of a base-structure where the piezoelectric device is attached. In this way, the applications can be broadened to other strain-based sensors.

The deformation of the structure can also be used to preserve the shape of a certain region of the structure, as showed by Zhu et al. (2016). The authors explored the minimization of the warping deformation to achieve a shape preserving effect. The same idea was applied to control the directional deformation behavior of the prescribed base-structure by Li et al. (2017). Very recently, Castro et al. (2018) carried out shape preserving design to minimized local deformation of vibrating structures. In other cases, prescribed displacements, deformation or motion are desired, such as in compliant mechanisms (Stanford and Beran 2012; Kim and Kim 2014; Zuo and Xie 2014). These works are all carried out under a density-based topology optimization approach.

This paper aims to develop a methodology to manipulate stresses and strains via level set topology optimization (LSTO). The shape of the structural boundaries are described via an implicit signed distance function and a fixed grid finite element analysis is used to evaluate the structural displacement field. A local von Mises stress measure can be used to minimize or maximize stresses in a prescribed sub-structure. The minimization of stress is applied as a stress isolation technique and its maximization as a design for failure procedure. A strain integral is introduced in order to maximize or minimize strains in a prescribed direction. A shape sensitivity analysis valid for both stress optimization and strain control is presented. We extend our previous work on stress-based topology optimization to achieve this (Picelli et al. 2018).

The remainder of the paper is organized as follows. Section 2 presents the structural description via a level set function and the finite element method. Section 3 describes the optimization formulation applied to stress and strain control. Section 4 briefly presents the level set topology optimization method. Section 5 shows numerical results and discussions and Section 6 concludes the paper.

## 2 Level set description and finite element method

*n*= 2, 3) occupied by a linear elastic isotropic structure defined by the domain \({\Omega }^{S}\). The structure is composed by a smooth boundary \(\partial {\Omega }^{S}\) = \({\Gamma }_{D} \cup {\Gamma }_{N} \cup {\Gamma }_{H}\). Dirichlet boundary conditions are applied in \({\Gamma }_{D}\), while homogeneous Neumann conditions are applied in \({\Gamma }_{N}\). The free boundaries are defined as \({\Gamma }_{H}\). Herein, \({\Gamma }_{H}\) is divided in two different sets, the outer boundaries \({\Gamma }_{H_{0}}\) and the inner boundaries \({\Gamma }_{H_{0}}^{\ast }\) from the holes of the structures, see Fig. 2a. If the set of boundaries \({\Gamma }_{H_{0}}^{\ast }\) is allowed to change whilst keeping \({\Gamma }_{H_{0}}\) fixed, only the inner holes are subjected to optimization.

## 3 Problem formulation and sensitivity analysis

*xx*direction, \(\boldsymbol {\varepsilon }_{xx} = \boldsymbol {\alpha } \cdot \boldsymbol {\varepsilon } \left (\mathbf {u} \right ) = \left (1, 0, 0\right )^{T} \cdot \boldsymbol {\varepsilon } \left (\mathbf {u} \right )\). Both stress and strain functions are integrals over the sub-structure control region \({\Omega }^{+}\).

As the results of our previous stress-based work suggest (Picelli et al. 2018), no regularization techniques (e.g. length scale or perimeter control) are needed in order to obtain smooth structural boundaries. However, in this work, a perimeter constraint is used in order to ensure the problem is well-posed, since volume is not constrained here. Differently from stress, the perimeter sensitivities are computed here via finite differences. One can efficiently compute perimeter sensitivities by locally checking the differences in the length of each point segment with a small perturbation of each boundary point coordinate in the normal direction. Our computational experience showed that the time for computing perimeter sensitivities is negligible if compared to solving the FEA equation using our open source code available at http://m2do.ucsd.edu/software/.

### 3.1 Computational procedure

*p*’s in the finite element mesh as

We employ isoparametric bilinear quadrilateral elements and stress is computed at four Gauss points per element. Although these elements present only one superconvergent point (the central one), the convergence properties of the integration points are still suggested to be used for sampling when the isoparametric element is not distorted (Zienkiewicz and Taylor 2005), which is the case of this paper. Based on the stress field at the Gauss points, the stresses at the boundary points are interpolated using the least squares method. This approach has been demonstrated in the context of both stress minimization and stress constrained level set topology optimization in our previous publication (Picelli et al. 2018).

## 4 Level set topology optimization

*k*is the iteration number,

*j*is a discrete boundary point, \({\Delta } t\) is the time step and \(V_{n}\) is the velocity at point

*j*, considered as an advection velocity of the type \(dx/dt = V_{n} \cdot \nabla \phi (x )\) (Osher and Fedkiw 2003). The velocities required for the level set update at every

*k*-th iteration are obtained by solving the following discretized optimization problem:

*f*and

*i*-th constraint functions \(g_{i}\). The terms \(s_{f}\) and \(s_{g_{i}}\) are the shape sensitivity functions for the objective and constraint functions, respectively, and \(l_{j}\) is the discrete length of the boundary associated with point

*j*.

The optimization problem from (22) is solved at every iteration *k*. The optimal velocities are then substituted into (21) to update the level-set boundaries. The process is repeated until the objective function stops changing during five consecutive iterations under a certain relative tolerance of \(10^{-3}\).

## 5 Numerical results

In this section numerical results and discussions are presented. First, the investigation of both stress and strain objectives is carried out for a square plane domain and optimization of the inner boundaries \({\Gamma }_{H_{0}}^{\ast }\). The method is then applied to stress isolation and stress maximization in a cantilever beam and directional strain control. In all examples, the structural perimeter is the only constraint applied.

### 5.1 Plane domain

*E*of the solid material is considered to be 200× 10

^{9}and the Poisson’s ratio \(\mu = 0.3\). The Young’s modulus of the void material is 10

^{− 9}.

The domain is discretized with 200×200 finite elements and two loading scenarios are considered, namely an uni-directional in-plane tension, Fig. 4a, and a bi-directional in-plane tension, Fig. 4b. The distributed force is 100. In this example, only the inner boundaries \({\Gamma }_{H_{0}}^{\ast }\) are subject to optimization.

#### 5.1.1 Stress isolation

*%*. The perimeter constraint is satisfied in the end of the optimization, as seen in Fig. 7c. Figure 8 presents the upper left quadrant of the same stress isolation solutions but starting with different initial holes and mesh sizes and all cases using the perimeter constraint of 200. The final topologies are similar, showing the perimeter control aids in reducing mesh dependency. The bi-directional in-plane tension case has a trivial solution when it comes to stress isolation, a circular hole around \({\Omega }^{+}\) which disconnects the base and sub-structure, case omitted herein.

*%*and 33

*%*, respectively.

#### 5.1.2 Strain control

^{− 7}. The minimization of \(\boldsymbol {\varepsilon }_{xx}\) leads to a mostly compressive strain field with only a few outer elements with positive

*xx*strain, being the total integral valued \(\boldsymbol {\varepsilon }_{xx}\) = -1.147× 10

^{− 7}. The solution is presented in Fig. 11a. It can be seen that the application of the tension load in the optimized structure induces compression in \({\Omega }^{+}\) region, see Fig. 11b. Consequently, the sub-structure becomes under tension in the

*yy*direction and \(\boldsymbol {\varepsilon }_{yy}\) becomes positive, as seen in Fig. 11c. Similar effects are seen in the uni-directional in-plane tension case of which the optimum solution is when \(\boldsymbol {\varepsilon }_{yy}\) is maximized, presented in Fig. 12.

^{− 5}.

^{− 7}to -1.967× 10

^{− 7}, implying that the sub-structure \({\Omega }^{+}\) becomes mostly under compression, predominantly in

*yy*direction.

^{o}.

Compilation of four different cases of strain control optimization

In-plane loading case | Objective function (OF) | Initial OF | Final OF | N | Optimum solution |
---|---|---|---|---|---|

uni-directional | min. \(\boldsymbol {\varepsilon }_{xx}+\boldsymbol {\varepsilon }_{yy}+\boldsymbol {\varepsilon }_{xy}\) | 4.894× 10 | -2.249× 10 | 178 | |

uni-directional | max. \(\boldsymbol {\varepsilon }_{xx}+\boldsymbol {\varepsilon }_{yy}+\boldsymbol {\varepsilon }_{xy}\) | 4.894× 10 | 2.001× 10 | 537 | |

bi-directional | max. \(\boldsymbol {\varepsilon }_{xx}\) | 1.118× 10 | 5.966× 10 | 467 | |

bi-directional | max. \(\boldsymbol {\varepsilon }_{xx} + \boldsymbol {\varepsilon }_{yy}\) | 2.798× 10 | 4.382× 10 | 287 |

### 5.2 Shape preserving and design for failure

This example shows the application of the proposed stress objective function to shape preserving and to design for failure. The minimization of deformation preserves the shape of a prescribed sub-structure \({\Omega }^{+}\) under the loading condition. For instance, Zhu et al. (2016) used a square measure of the strain in order to minimize the deformation energy in the sub-structure. In this work, this is achieved by stress isolation. The case of design for failure is the opposite. In such case, the stress in \({\Omega }^{+}\) should be maximized in order to ensure the structure will fail in the prescribed region, preserving the integrity of other parts.

^{9}and the Poisson’s ratio \(\mu = 0.3\). Figure 16 shows the initial stress field and the initial solution used in the level set optimization problem.

*%*if compared with the value in the initial structure (4779.12). Figure 17c–d presents the solution for stress integral maximization in \({\Omega }^{+}\). The maximum stress point inside \({\Omega }^{+}\) ensures structural failure in that region or in its vicinity. The final integral on \({\Omega }^{+}\) for stress maximization is 97964.10, an increase of \(\approx \) 1950

*%*.

^{8}.

### 5.3 Directional strain control

^{9}and the Poisson’s ratio \(\mu = 0.3\). The perimeter is constrained to 500. Figure 20 shows the solutions for \(\boldsymbol {\varepsilon }_{xx}\) and \(\boldsymbol {\varepsilon }_{yy}\) minimization, i.e., using \(\alpha = \left \{1, 0, 0\right \}\) and \(\alpha = \left \{0, 1, 0\right \}\), respectively. Notice that we are solving a stress isolation case, but with an unity value in the elasticity matrix to select the strain direction of interest. The same initial solution from Fig. 16a is used for this example.

*x*direction and compressed in the

*y*direction. The minimization of stress using \(\alpha \) to select a prescribed direction leads to the minimization of deformation in that direction. Figure 22 presents the deformed \({\Omega }^{+}\) in the initial structure and for \(\boldsymbol {\varepsilon }_{xx}\) and \(\boldsymbol {\varepsilon }_{yy}\) minimization. The plotted displacements are scaled by 10

^{8}.

## 6 Conclusions

This paper presented a level set optimization method for stress and strain manipulation. The shape and topology modification of a base-structure \({\Omega }^{S}\) allowed the control of stress flux inside a sub-structure \({\Omega }^{+}\). A general integral objective function was proposed and the sensitivities were derived. For stress isolation, the von Mises measure was used. Numerical results show that stresses in \({\Omega }^{+}\) can be efficiently isolated in the presented examples via the optimization of the hole shapes in the base-structure, achieving the drastic reduction as much as 89*%* from the initial stress integral. The increase in the stress flux on the base-structure was also achieved by maximization of the stress integral. This may be applied when a failure needs to be designed in for prognosis. An increase of 126*%* was achieved in the bi-directional in-plane tension case.

It was shown that the solutions for strain control can be considerably different from those obtained for stress optimization. A strain objective function was proposed based on a vector \(\boldsymbol {\alpha }\) able to select the strain component of interest. It was found that optimization explores the directionality of strain in the optimum solution, e.g., minimization of strain achieves a negative strain integral starting from a positive initial value (tension). A few other examples were briefly compiled.

The stress objective showed that it can also be used for shape preserving design and directional strain control. This paper therefore presents a level set optimization method that can manipulate stress and/or strain in a specified sub-structure. This can be used for stress isolation of highly sensitive non strain-based sensors, design for failure, maximization of mechanical strain, strain direction control and shape preserving design.

## Notes

### Acknowledgements

We thank the support of the Engineering and Physical Sciences Research Council, fellowship grant EP/M002322/2. The authors would also like to thank the Numerical Analysis Group at the Rutherford Appleton Laboratory for their FORTRAN HSL packages (HSL, a collection of Fortran codes for large-scale scientific computation. See http://www.hsl.rl.ac.uk/).

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