Stress and strain control via level set topology optimization

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.

Stress isolation aims to minimize the stress levels in prescribed regions of a structure. This can be achieved by redirecting and attenuating the propagation of stresses from a base-structure \({\Omega }^{S}\) to a sub-structure \({\Omega }^{+}\) with the use of different materials or adding holes or cuts, see Fig. 1. In this sense, typical applications are in the design of sensors that are largely sensitive to mechanical stress, such as acceleration sensors (Tsuchiya and Funabashi 2004; Hsieh et al. 2011). Another example is the structural design for welding stress isolation, as patented by Gorard et al. (2011). In this case, the source of stress is in the sub-structure and the aim is to eliminate stress propagation to the base-structure.

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.

Let \({\Omega }\) be a bounded design domain in \(\text {I\!R}^{n}\) (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.

In level set topology optimization, the structural boundaries are represented by an implicit function as where \({\Phi }\) is the level set function, \(\mathrm {x}\) is the point inside the design domain \({\Omega }\) and \({\Omega }^{S} \subset {\Omega }\) (Allaire et al. 2004). In this work, the level set is defined as a signed distance function, as seen in Fig. 2b.

The analysis is conducted by the fixed grid finite element method with equivalent Ersatz material. In this approach, the domain \({\Omega }^{S}\) is projected onto a fixed grid that covers \({\Omega }\) and the boundary \({\Gamma }_{H}\) is discretized into a set of points coincident with the elements edges, as illustrated in Fig. 3. Cut elements are assigned with volume fractions between 0 and 1, according to the ratio of volume of solid material and the total volume of the element (Dunning et al. 2011).

The structure \({\Omega }\) is in equilibrium when a displacement field \(\mathbf {u}\) satisfies the equation, where the linear operators \(a\left (\cdot ,\cdot \right )\) and \(l\left (\cdot \right )\) represent the virtual work of internal and external forces, respectively, where \(\boldsymbol {\sigma }\left (\mathbf {u} \right )\) and \(\boldsymbol {\varepsilon }\left (\mathbf {v} \right )\) are the stress and strain vectors, respectively, \(\mathbf {t}\) is a constant and non-zero traction on \({\Gamma }_{N}\) and \(\mathbf {v}\) is the field of virtual variations. No body forces are considered.

The stress objective and strain objective functions are defined, respectively, as where \(\sigma _{vm}\) is the von Mises stress of the structure, \(\boldsymbol {\varepsilon } \left (\mathbf {u}\right )\) is the strain with all components \(\boldsymbol {\varepsilon }_{xx}\), \(\boldsymbol {\varepsilon }_{yy}\) and \(\boldsymbol {\varepsilon }_{xy}\) and \(\boldsymbol {\alpha }\) is a vector to select the strain direction, when desired. For example, for the strain in the 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 \({\Omega }^{+}\).

The level set topology optimization requires the computation of shape sensitivities at the boundary points from Fig. 3a in order to update the implicit level set function. The shape sensitivity of a general function \(f\left ({\Omega } \right )\) can be derived using the adjoint method. This technique is advantageous when using a large number of design variables, the typical case of topology optimization. An augmented integral function can be expressed with the aid of the equilibrium equation where \(\boldsymbol {\lambda }\) is the adjoint variable. The material derivative method is applied here (Wang and Li 2013; Choi and Kim 2005) and the derivative of the augmented functional \(\mathcal {L}\) can be expressed as where, in which \(V_{n}\) is the normal velocity component of a boundary \({\Gamma }\) and \(^{\prime }\) indicates the derivative with respect to this boundary.

Substituting (7), (8) and (9) into (6) and collecting all the terms that contain \(\boldsymbol {\lambda }^{\prime }\), the weak form of the state equation is recovered, whose total sum is zero. By collecting all the terms that depend on \(\mathbf {u}^{\prime }\) and letting their sum be zero, the following adjoint equation is obtained where the right-hand side of (10) indicates the adjoint pseudo-load. Finally, collecting all the remaining terms, the shape derivative of the augmented functional with respect to the boundary points on \({\Gamma }\) can be written as

For the stress objective, the pseudo-load from (10) can be expressed as where \(H\left (\mathrm {x} \right )\) is the Heaviside function By using the Heaviside function, the pseudo-load is in the form of the general shape derivative and can be used to solve the adjoint problem from (10) and obtain \(\boldsymbol {\lambda }\). For the strain objective, the pseudo-load is written as

Substituting the objective function \(f^{+}\left (\mathbf {u}; \mathrm {x} \right )\) in (11), one can rewrite it in terms of the boundary \({\Gamma }_{H_{0}}^{\ast }\) to be optimized Both stress and strain objectives are defined within the space of \({\Omega }^{+}\). In this work, \({\Omega }^{+}\) is considered as a non-design domain to represent the sub-structure location of a sensor, thus, zero velocities are assigned in those points. Therefore, the first term of (15) is null and the shape sensitivities at the boundary points can be computed as For stress isolation, \(\boldsymbol {\lambda }\) is obtained by solving the adjoint problem from (10) with the pseudo-load from (12), while for strain control the pseudo-load is computed as (14). The derivatives in both integrals from (12) and (14) are computed with respect to the displacement field.

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/​.

The level set topology optimization method used follows Picelli et al. (2018) and it is briefly described here for completeness. The structural boundary is optimized by updating the implicit level set function via an advection equation combined with the velocity field of motion. The discretized version of such equation can be expressed as where 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: where \(\bar {g}^{k}_{i}\) is a relaxed change in the constraint function at every iteration and the velocities \(\mathbf {V}^{k}_{n}\) are described in terms of a search direction \(\mathbf {d}\) normal to the boundary and the actual distance \(\beta > 0\), as The Lagrangian function related to the problem given in (22) is expressed as where \(\lambda ^{k}\) is a Lagrange multiplier and \(\mathbf {S}_{f}^{k}\) and \(\mathbf {S}_{i}^{k}\) are vectors containing integral coefficients computed with the shape sensitivities at the boundary points as for linearized objective 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 \(10^{-3}\).

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.

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.