Pixel-based shape optimization in 3D using constrained density-based topology optimization

Structural optimization methods, such as shape and topology optimization, hold significant promise for enhancing the design of acoustic metamaterials. Prior research emphasizes the necessity of incorporating viscous and thermal losses into the optimization process for resonator-based acoustic metamaterials. While acoustic losses can be integrated into boundary element and finite element frameworks, shape optimization often lacks the design flexibility required for complex metamaterial structures. Consequently, topology optimization is frequently preferred due to its greater design freedom. Resent research has demonstrated that density-based topology optimization can include acoustic losses using an indicator function combined with the boundary layer impedance condition. As an alternative to density-based topology optimization, this presentation will discuss implementation aspects of a cut-elements and level-set based topology optimization approach including viscous and thermal losses. The cut-elements method offers advantages over traditional density-based approaches by providing a clear interface between acoustic and rigid domains, ensuring that each design iteration accurately reflects the underlying physical behavior.

This paper proposes a multi-stage design method for a design of practical compliant mechanisms. The proposed method consists of topology and shape optimizations and a shape conversion method that incorporates two optimizations. In the 1st stage, an initial and conceptual compliant mechanism is created by topology optimization. In the 2nd stage, an initial model of shape optimization is created from the result of topology optimization by the shape conversion method based on the level set method. In the 3rd stage, the shape optimization yields a detailed shape of the compliant mechanism by considering non-linear deformation and stress concentration. Execution of the shape optimization after the topology optimization enables evaluation of stress concentration and large deformation effect that are normally difficult for the traditional topology optimization. On the other side, the precise conversion from the model by topology optimization to the one for the shape optimization becomes possible by the shape conversion method that is utilizing the level set method. Using the proposed multi-stage method, a practical compliant mechanism can be designed with the designer’s minimum efforts that are indications of design conditions of the topology and shape optimizations and several parameters and threshold values of the shape conversion method.

This thesis contains contributions to the development of topology optimization techniques capable of handling stress constraints. The research that led to these contributions was motivated by the need for topology optimization techniques more suitable for industrial applications. Currently, topology optimization is mainly used in the initial design phase, and local failure criteria such as stress constraints are considered in additional post-processing steps. Consequently, there is often a large gap between the topology optimized design and the final design for manufacturing. Taking into account stress constraints directly into the topology optimization process would reduce this gap. Several difficulties arise in topology optimization with local stress constraints which complicate solving the optimization problem directly. \chap{litreview} discusses these difficulties, and reviews solutions that have been applied. Two fundamental difficulties are: (i) the presence of singular optima, which are true optima inaccessible to standard nonlinear programming techniques, and (ii) the fact that the stress is a local state variable, which typically leads to a large number of constraints. Currently, the conventional strategy to circumvent these difficulties is to apply (i) constraint relaxation, which perturbs the feasible domain to make singular optima accessible, followed by (ii) constraint aggregation to transform the typically large number of relaxed constraints into a single or few global constraints thereby reducing the order of the problem. Although there is no consensus on the exact choice of aggregation and relaxation functions and their numerical implementation, in general, this approach introduces two additional parameters to the problem: an aggregation and a relaxation parameter. Following this approach, one solves an alternative optimization problem with the aim of finding a solution to the original stress-constrained topology optimization. The feasible domain of this alternative optimization problem is related to the original feasible domain via these parameters. In Chapter 2, we investigated the parameter dependence of this alternative optimization problem on an elementary two-bar truss problem. It was found that the location of the global optimum of this alternative optimization problem with respect to the true optimum depends in a non-trivial way on these problem parameters (in their range of application); i.e., for a given parameter set, it is difficult to predict the influence of changing one of the parameter values, and if this change will result in a feasible domain in which the global optimum is closer to the true optimum. This complicates determining optimal parameter values \emph{a priori} which, in addition, are problem-dependent. In Chapter 3, we investigated the effect of design parameterization, and relaxation techniques in stress-constrained topology optimization. An elementary numerical example was considered, representing a situation as might occur in density-based topology optimization. As previously observed in truss optimization, we found that a global optimum of the relaxed optimization problem may not converge to the true optimum as the relaxation parameter is decreased to zero. In this thesis, we present two novel approaches: a unified aggregation and relaxation approach in Chapter 4, and the damage approach in Chapter 5. In the unified aggregation and relaxation approach, we applied constraint aggregation such that it simultaneously perturbs the feasible domain, and makes singular optima accessible. Consequently, conventional relaxation techniques become unnecessary when applying constraint aggregation following this approach. The main advantage is that the problem only depends on a single parameter, which reduces the parameter dependency of the problem. The damage approach is presented as a viable alternative for conventional methodologies. Following the damage approach stress constraint violation is penalized by degrading material where the stress exceeds the allowable stress. Material degradation affects the overall performance of the structure, and therefore, the optimizer promotes a design without stress constraint violation. Similar to conventional constraint aggregation techniques a large number of local constraints can be controlled by imposing a single or a few global constraints. Both novel approaches are validated on elementary truss examples and tested on numerical examples in density-based topology optimization. In contrast to the conventional strategy of relaxation followed by aggregation, there exists a clear relationship between the perturbed feasible domain and the original unperturbed feasible domain in terms of a single problem parameter.

In order to generate a reliable design the nonlinear structural response, e.g. buckling or plasticity, has to be considered in topology and shape optimization. In the present study material topology optimization determining the basic layout is extended to elastoplasticity. Afterwards the shape of the boundaries is optimized by shape optimization also considering the nonlinear material behavior. An elastoplastic von Mises material model with linear, isotropic hardening/softening for small strains is used. The objective of the design problem is to maximize the structural ductility defined by the strain energy over a given range of a prescribed displacement. With respect to the specific features of topology and shape optimization, e.g. the number of optimization variables or local-global influence of optimization variables on the structural response, different numerical methods are applied to solve the respective optimization problem. In topology optimization the gradient of the ductility is determined by the variational adjoint approach. In shape optimization the derivatives of the state variables with respect to the optimization variables are evaluated analytically by a variational direct approach. Topology optimization problems are solved by optimality criteria (OC) methods, shape optimization problems by mathematical programming (MP) methods, i.e. SQP-algorithm. In topology optimization a geometrically adaptive optimization procedure is additionally applied in order to increase the numerical efficiency and to avoid artificial stress singularities. The numerical procedures are verified by a 2D design problem under plane stress conditions.

Density-based topology optimization and node-based shape optimization are often used sequentially to generate production-ready designs. In this work, we address the challenge to couple density-based topology optimization and node-based shape optimization into a single optimization problem by using an embedding domain discretization technique. In our approach, a variable shape is explicitly represented by the boundary of an embedded body. Furthermore, the embedding domain in form of a structured mesh allows us to introduce a variable, pseudo-density field. In this way, we attempt to bring the advantages of both topology and shape optimization methods together and to provide an efficient way to design fine-tuned structures without predefined topological features.

Body-fitted topology optimization of 2D and 3D fluid-to-fluid heat exchangers

In the present paper, a model for growth of elastic bodies is proposed for purposes of shape and topology optimization. Growth of solid bodies is herewith considered in the framework of topology and shape optimization, with the goal of mimicking the natural generation of both stiff and light biological structures, consisting of a solid skeleton immersed into a softer phase. The growth process is modeled as the nucleation and subsequent growth of islands of a hard elastic phase within a softer elastic matrix, which plays the role of a reservoir of nutrients for the supply of the newly generated material. Islands of the generated solid skeleton are modeled as balls of small radius, and the position of their center is determined in such a way that the effective compliance, the product of the compliance by the relative density of the hard phase, is minimal for each new generation event. A growth model is set up from the mass balance with a source term involving the growth rate of mass, taken as a constant. The modeling of the growth process relies on the evaluation of the topological derivative of the effective compliance, which allows finding the optimal position of the center of new inclusions of the generated hard phase. This nucleation process is then followed by the shape optimization of the growing solid bodies. A proper mathematical formulation of the topology and shape optimization of growing elastic solid body is provided, relying on a domain decomposition technique allowing to replace the singularly perturbed geometrical domain by a regularly perturbation of Steklov-Poincaré operator. As an enrichment of this model, surface energy is lastly considered in the framework of a linear elastic constitutive model with surface stress.

Shape optimization of cold-formed steel columns with fabrication and geometric end-use constraints

Compliant mechanisms designed by traditional topology optimization have a linear output response, and it is difficult for traditional methods to implement mechanisms having nonlinear output responses, such as nonlinear deformation or path. To design a compliant mechanism having a specified nonlinear output path, we propose a two-stage design method based on topology and shape optimizations. In the first stage, topology optimization generates an initial conceptual compliant mechanism based on ordinary design conditions, with “additional” constraints used to control the output path in the second stage. In the second stage, an initial model for the shape optimization is created, based on the result of the topology optimization, and additional constraints are replaced by spring elements. The shape optimization is then executed, to generate the detailed shape of the compliant mechanism having the desired output path. At this stage, parameters that represent the outer shape of the compliant mechanism and of spring element properties are used as design variables in the shape optimization. In addition to configuring the specified output path, executing the shape optimization after the topology optimization also makes it possible to consider the stress concentration and large displacement effects. This is an advantage offered by the proposed method, because it is difficult for traditional methods to consider these aspects, due to inherent limitations of topology optimization.

Topology optimization has become very popular in industrial applications, and most FEM codes have implemented certain capabilities of topology optimization. However, most codes do not allow simultaneous treatment of sizing and shape optimization during the topology optimization phase. This poses a limitation on the design space and therefore prevents finding possible better designs since the interaction of sizing and shape variables with topology modification is excluded. In this paper, an integrated approach is developed to provide the user with the freedom of combining sizing, shape, and topology optimization in a single process.

Density-based topology optimization integrated with genetic algorithm for optimizing formability and bending stiffness of 3D printed CFRP core sandwich sheets

For aeronautical applications of topology optimization, it is of importance to develop topology optimization techniques, that can handle stress constraints in an efficient and accurate manner. The development of such topology optimization techniques is a challenging task due to the local nature of the stress constraints, their highly non-linear behaviour with respect to the design variables and the so-called singularity phenomenon. An accurate sensitivity analysis is essential for these type of problems with multiple constraints. In this paper, we propose a methodology of dealing with stress constraints in a level set based framework. In this framework, the level set function nodal values are related to element densities by an exact Heaviside projection. Stress relaxation and constraint aggregation techniques are used to deal with the singularity phenomenon and the local nature of the stress, respectively. A constrained optimization problem is then solved, in which the design variables (the level set nodal values) are updated in the projected steepest-descent direction, which is determined using a consistent sensitivity analysis.We demonstrate the effectiveness of this technique on two numerical examples. The results show that the level set method with a consistent sensitivity analysis allows for the treatment of multiple constraints by using constrained optimization techniques.

Density-based topology optimization using global and local volume constraints is a key technique to automatically design lightweight structures. It is known that stiffness optimal structures comprise spatially varying geometric patterns that span multiple length scales. However, both variants of topology optimization have challenges to efficiently converge to such a structural layout. In this paper, we investigate material layouts that are generated from stress trajectories, i.e., to compile a globally consistent structure by tracing the stress trajectories from finite element simulation of the solid design domain under external loads. This is particularly appealing from a computational perspective, since it avoids iterative optimization that involves finite element analysis on fine meshes. By regularizing the thickness of each trajectory using derived strain energy measures along them, stiff structural layouts can be generated in a highly efficient way. We then shed light on the use of the resulting structures as initial density fields in density-based topology optimization, i.e., to generate an initial density field that is then further optimized via topology optimization. We demonstrate that by using a stress trajectory guided density initialization in lieu of a uniform density field, convergence issues in density-based topology optimization can be significantly relaxed at comparable stiffness of the resulting structural layouts.

This paper was concerned with a new integrated method of topology optimization and shape optimization. The new method was used in this paper to optimize the structure of an air suspension bracket. In this article, Method of Moving Asymptotes was employed to solve the variable density model discrete by finite element method. The compliance and mass were regarded as topology and shape optimization objectives, respectively. In topology optimization, filter functions of elements with respect to the unit volume and stiffness matrices were selected based on Solid Isotropic Material with Penalization (SIMP) interpolation scheme. Based on the results of the topology optimization, the conceptual model was design, but the max stress was larger than the original model and the stress distribution was not even. Shape optimization was introduced to the modified model. In shape optimization, efficient derivatives were obtained using approximate-difference strategy. Structural volume was taken as objectives and structural response stresses were taken as constraints which ensure consistency in topology optimization and shape optimization models. Volume of the air suspension bracket was minimized subjected to rigidity and strength constraints. Finally, the detailed bracket structure is designed based on the results of the optimization. Compared with the traditional structure, the weight of the new optimized structure is reduced 30 percent and the structural reliability is almost the same with the traditional structure.

A sequentially coupled shape and topology optimization framework is presented in which the outer geometry and the internal topological layout of beam-type structures are optimized simultaneously. The outer geometry of the beam-type structures is parametrically described by non-uniform rational B-splines (NURBS), which guarantees a highly accurate description of the structural shape and enable an efficient control of the design domain with only a few control points. The computational efficiency of the coupled optimization approach is assured by applying a gradient-based optimization algorithm, for which the sensitivities are derived in closed form. The formulation of the coupled optimization approach is tailored toward 2.5D and full 3D representations of beam structures used in engineering applications. The 2.5D beam model, which has been taken from the literature, uses standard beam elements to simulate the beam response in the longitudinal direction, whereby the cross-sectional properties of the beam elements are calculated from additional 2D finite element method (FEM) analyses. A comparison study of a cantilever beam problem subjected to pure shape optimization and pure topology optimization illustrates that the 2.5D and 3D beam models lead to similar shape and topology designs, but that the 2.5D beam model has a significantly higher computational efficiency. Specifically, the computational times for the 2.5D model are about a factor 70 (shape optimization) and 1.4 (topology optimization) lower than for the 3D model, which indicates that in the coupled optimization approach the optimization of the shape provides the largest contribution to the higher computational efficiency of the 2.5D model. The coupled shape and topology optimization analysis subsequently performed on the 2.5D cantilever beam model demonstrates that the specific order at which the alternating shape and topology optimization increments are performed in the staggered update procedure turns out to have some influence on the computational speed and the value of the minimal compliance computed. Despite these differences, the final beam structures following from the different staggered update procedures illustrate how shape and topology can be efficiently optimized in an integrated, coupled fashion.