A level set‐based topology optimization method targeting metallic waveguide design problems

In this paper, we propose a level set‐based topology optimization method targeting metallic waveguide design problems, where the skin effect must be taken into account since the metallic waveguides are generally used in the high‐frequency range where this effect critically affects performance. One of the most reasonable approaches to represent the skin effect is to impose an electric field constraint condition on the surface of the metal. To implement this approach, we develop a boundary‐tracking scheme for the arbitrary Lagrangian Eulerian (ALE) mesh pertaining to the zero iso‐contour of the level set function that is given in an Eulerian mesh, and impose Dirichlet boundary conditions at the nodes on the zero iso‐contour in the ALE mesh to compute the electric field. Since the ALE mesh accurately tracks the zero iso‐contour at every optimization iteration, the electric field is always appropriately computed during optimization. For the sensitivity analysis, we compute the nodal coordinate sensitivities in the ALE mesh and smooth them by solving a Helmholtz‐type partial differential equation. The obtained smoothed sensitivities are used to compute the normal velocity in the level set equation that is solved using the Eulerian mesh, and the level set function is updated based on the computed normal velocity. Finally, the utility of the proposed method is discussed through several numerical examples. Copyright © 2011 John Wiley & Sons, Ltd.     

A moving superimposed finite element method for structural topology optimization

Level set methods are becoming an attractive design tool in shape and topology optimization for obtaining efficient and lighter structures. In this paper, a dynamic implicit boundary‐based moving superimposed finite element method (s‐version FEM or S‐FEM) is developed for structural topology optimization using the level set methods, in which the variational interior and exterior boundaries are represented by the zero level set. Both a global mesh and an overlaying local mesh are integrated into the moving S‐FEM analysis model. A relatively coarse fixed Eulerian mesh consisting of bilinear rectangular elements is used as a global mesh. The local mesh consisting of flexible linear triangular elements is constructed to match the dynamic implicit boundary captured from nodal values of the implicit level set function. In numerical integration using the Gauss quadrature rule, the practical difficulty due to the discontinuities is overcome by the coincidence of the global and local meshes. A double mapping technique is developed to perform the numerical integration for the global and coupling matrices of the overlapped elements with two different co‐ordinate systems. An element killing strategy is presented to reduce the total number of degrees of freedom to improve the computational efficiency. A simple constraint handling approach is proposed to perform minimum compliance design with a volume constraint. A physically meaningful and numerically efficient velocity extension method is developed to avoid the complicated PDE solving procedure. The proposed moving S‐FEM is applied to structural topology optimization using the level set methods as an effective tool for the numerical analysis of the linear elasticity topology optimization problems. For the classical elasticity problems in the literature, the present S‐FEM can achieve numerical results in good agreement with those from the theoretical solutions and/or numerical results from the standard FEM. For the minimum compliance topology optimization problems in structural optimization, the present approach significantly outperforms the well‐recognized ‘ersatz material’ approach as expected in the accuracy of the strain field, numerical stability, and representation fidelity at the expense of increased computational time. It is also shown that the present approach is able to produce structures near the theoretical optimum. It is suggested that the present S‐FEM can be a promising tool for shape and topology optimization using the level set methods. Copyright © 2005 John Wiley & Sons, Ltd.     

Adaptive moving mesh level set method for structure topology optimization

The level set method is a promising approach to provide flexibility in dealing with topological changes during structural optimization. Normally, the level set surface, which depicts a structure's topology by a level contour set of a continuous scalar function embedded in space, is interpolated on a fixed mesh. The accuracy of the boundary positions is therefore largely dependent on the mesh density, a characteristic of any Eulerian expression when using a fixed mesh. This article combines the adaptive moving mesh method with a level set structure topology optimization method. The finite element mesh automatically maintains a high nodal density around the structural boundaries of the material domain, whereas the mesh topology remains unchanged. Numerical experiments demonstrate the effect of the combination of a Lagrangian expression for a moving mesh and a Eulerian expression for capturing the moving boundaries.     

Three‐dimensional grayscale‐free topology optimization using a level‐set based r‐refinement method

In this paper, we propose a three‐dimensional (3D) grayscale‐free topology optimization method using a conforming mesh to the structural boundary, which is represented by the level‐set method. The conforming mesh is generated in an r‐refinement manner; that is, it is generated by moving the nodes of the Eulerian mesh that maintains the level‐set function. Although the r‐refinement approach for the conforming mesh generation has many benefits from an implementation aspect, it has been considered as a difficult task to stably generate 3D conforming meshes in the r‐refinement manner. To resolve this task, we propose a new level‐set based r‐refinement method. Its main novelty is a procedure for minimizing the number of the collapsed elements whose nodes are moved to the structural boundary in the conforming mesh; in addition, we propose a new procedure for improving the quality of the conforming mesh, which is inspired by Laplacian smoothing. Because of these novelties, the proposed r‐refinement method can generate 3D conforming meshes at a satisfactory level, and 3D grayscale‐free topology optimization is realized. The usefulness of the proposed 3D grayscale‐free topology optimization method is confirmed through several numerical examples. Copyright © 2017 John Wiley & Sons, Ltd.     

Structural shape and topology optimization using a meshless Galerkin level set method

This paper aims to propose a meshless Galerkin level set method for shape and topology optimization of continuum structures. To take advantage of the implicit free boundary representation scheme, the design boundary is represented as the zero level set of a scalar level set function, to flexibly handle complex shape fidelity and topology changes by maintaining concise and smooth interface. Compactly supported radial basis functions (CSRBFs) are used to parameterize the level set function and construct the shape functions for meshfree approximations based on a set of unstructured field nodes. The meshless Galerkin method with global weak form is used to implement the discretization of the state equations. This provides a pathway to unify the two different numerical stages in most conventional level set methods: (1) the propagation of discrete level set function on a set of Eulerian grid and (2) the approximation of discrete equations on a set of Lagrangian mesh. The original more difficult shape and topology optimization based on the level set equation is transformed into a relatively easier size optimization, to which many efficient optimization algorithms can be applied. The proposed level set method can describe the moving boundaries without remeshing for discontinuities. The motion of the free boundary is just a question of advancing the discrete level set function in time by solving the size optimization. Several benchmark examples are used to demonstrate the effectiveness of the proposed method. The numerical results show that the proposed method can simplify numerical process and avoid numerical difficulties involved in most conventional level set methods. It is straightforward to apply the proposed method to more advanced shape and topology optimization problems. Copyright © 2011 John Wiley & Sons, Ltd.     

A consistent grayscale‐free topology optimization method using the level‐set method and zero‐level boundary tracking mesh

This paper proposes a level‐set based topology optimization method incorporating a boundary tracking mesh generating method and nonlinear programming. Because the boundary tracking mesh is always conformed to the structural boundary, good approximation to the boundary is maintained during optimization; therefore, structural design problems are solved completely without grayscale material. Previously, we introduced the boundary tracking mesh generating method into level‐set based topology optimization and updated the design variables by solving the level‐set equation. In order to adapt our previous method to general structural optimization frameworks, the incorporation of the method with nonlinear programming is investigated in this paper. To successfully incorporate nonlinear programming, the optimization problem is regularized using a double‐well potential. Furthermore, the sensitivities with respect to the design variables are strictly derived to maintain consistency in mathematical programming. We expect the investigation to open up a new class of grayscale‐free topology optimization. The usefulness of the proposed method is demonstrated using several numerical examples targeting two‐dimensional compliant mechanism and metallic waveguide design problems. Copyright © 2014 John Wiley & Sons, Ltd.     

Explicit level‐set‐based topology optimization using an exact Heaviside function and consistent sensitivity analysis

This paper presents a level‐set‐based topology optimization method based on numerically consistent sensitivity analysis. The proposed method uses a direct steepest‐descent update of the design variables in a level‐set method; the level‐set nodal values. An exact Heaviside formulation is used to relate the level‐set function to element densities. The level‐set function is not required to be a signed‐distance function, and reinitialization is not necessary. Using this approach, level‐set‐based topology optimization problems can be solved consistently and multiple constraints treated simultaneously. The proposed method leads to more insight in the nature of level‐set‐based topology optimization problems. The level‐set‐based design parametrization can describe gray areas and numerical hinges. Consistency causes results to contain these numerical artifacts. We demonstrate that alternative parameterizations, level‐set‐based or density‐based regularization can be used to avoid artifacts in the final results. The effectiveness of the proposed method is demonstrated using several benchmark problems. The capability to treat multiple constraints shows the potential of the method. Furthermore, due to the consistency, the optimizer can run into local minima; a fundamental difficulty of level‐set‐based topology optimization. More advanced optimization strategies and more efficient optimizers may increase the performance in the future. Copyright © 2012 John Wiley & Sons, Ltd.     

A velocity field level set method for shape and topology optimization

In this paper, we propose a new implementation of the level set shape and topology optimization, the velocity field level set method. Therein, the normal velocity field is constructed with specified basis functions and velocity design variables defined on a given set of points that are independent of the finite element mesh. A general mathematical programming algorithm can be employed to find the optimal normal velocities on the basis of the sensitivity analysis. As compared with conventional level set methods, mapping the variational boundary shape optimization problem into a finite‐dimensional design space and the use of a general optimizer makes it more efficient and straightforward to handle multiple constraints and additional design variables. Moreover, the level set function is updated by the Hamilton‐Jacobi equation using the normal velocity field; thus, the inherent merits of the implicit representation is retained. Therefore, this method combines the merits of both the general mathematical programming and conventional level set methods. Integrated topology optimization of structures with embedded components of designable geometries is considered to show the capability of this method to deal with general design variables. Several numerical examples in 2D or 3D design domains illustrate the robustness and efficiency of the method using different basis functions.     

A recursive application of the integral equation in the boundary element method

This paper presents a recursive application of the governing integral equation aimed at improving the accuracy of numerical results of the boundary element method (BEM). Usually, only the results at internal domain points when using BEM are found using this approach, since the nodal boundary values have already been calculated. Here, it is shown that the same idea can be used to obtain better accuracy for the boundary results as well. Instead of locating the new source points inside the domain, they are positioned on the boundary, with different coordinates to the nodal points. The procedure is certainly general, but will be presented using as an example the two dimensional Laplace equation, for the sake of simplicity to point out the main concepts and numerical aspects of the method proposed, especially due to the determination of directional derivatives of the primal variable, which is part in hyper-singular BEM theory.     

An adaptive mesh-adjustment strategy for continuum topology optimization to achieve manufacturable structural layout

This paper presents an adaptive mesh adjustment algorithm for continuum topology optimization method to describe the structural boundary using nonuniform isoparametric element. A criterion on the basis of the node movement is proposed; herein, the densities and coordinates of the nodes are defined to instruct the deformation of finite elements in subsequent optimization iterations. With such a scheme, the topology optimization can start from a regular mesh discretization then gradually yields an optimal design with clear structural boundaries. The element in the transition along the boundary is refined; on the contrary, the pure solid or void element is coarsen. The contribution of this work is to improve the resolution of the structural boundaries and decrease the percentage of transitional regions with the invariant design variable. Several 2D and 3D numerical examples indicate the effectiveness of our proposed method. Seen from the examples, the structural boundary become smoother and the intermediate densities have been reduced up to 70%. In addition, a design process based on the presented method is proposed to make the optimum solutions be fabricated conveniently and accurately by linking it with the 3D design software, ie, SolidWorks, which is also demonstrated in the numerical examples.     

Unification of parametric and implicit methods for shape sensitivity analysis and optimization with fixed mesh

Parametric and implicit methods are traditionally thought to be two irrelevant approaches in structural shape optimization. Parametric method works as a Lagrangian approach and often uses the parametric boundary representation (B‐rep) of curves/surfaces, for example, Bezier and B‐splines in combination with the conformal mesh of a finite element model, while implicit method relies upon level‐set functions, that is, implicit functions for B‐rep, and works as an Eulerian approach in combination with the fixed mesh within the scope of extended finite element method or finite cell method. The original contribution of this work is the unification of both methods. First, a new shape optimization method is proposed by combining the features of the parametric and implicit B‐reps. Shape changes of the structural boundary are governed by parametric B‐rep on the fixed mesh to maintain the merit in computer‐aided design modeling and avoid laborious remeshing. Second, analytical shape design sensitivity is formulated for the parametric B‐rep in the framework of fixed mesh of finite cell method by means of the Hamilton–Jacobi equation. Numerical examples are solved to illustrate the unified methodology. Copyright © 2016 John Wiley & Sons, Ltd.     

Acoustic shape optimization using cut finite elements

Fictitious domain methods are attractive for shape optimization applications, since they do not require deformed or regenerated meshes. A recently developed such method is the CutFEM approach, which allows crisp boundary representations and for which uniformly well‐conditioned system matrices can be guaranteed. Here, we investigate the use of the CutFEM approach for acoustic shape optimization, using as test problem the design of an acoustic horn for favorable impedance‐matching properties. The CutFEM approach is used to solve the Helmholtz equation, and the geometry of the horn is implicitly described by a level‐set function. To promote smooth algorithmic updates of the geometry, we propose to use the nodal values of the Laplacian of the level‐set function as design variables. This strategy also improves the algorithm's convergence rate, counteracts mesh dependence, and, in combination with Tikhonov regularization, controls small details in the optimized designs. An advantage with the proposed method is that the exact derivatives of the discrete objective function can be expressed as boundary integrals, as opposed to when using a traditional method that uses mesh deformations. The resulting horns possess excellent impedance‐matching properties and exhibit surprising subwavelength structures, not previously seen, which are possible to capture due to the fixed mesh approach.     

Topology optimization of structures using meshless density variable approximants

This paper proposes a new structural topology optimization method using a dual‐level point‐wise density approximant and the meshless Galerkin weak‐forms, totally based on a set of arbitrarily scattered field nodes to discretize the design domain. The moving least squares (MLS) method is used to construct shape functions with compactly supported weight functions, to achieve meshless approximations of system state equations. The MLS shape function with the zero‐order consistency will degenerate to the well‐known ‘Shepard function’, while the MLS shape function with the first‐order consistency refers to the widely studied ‘MLS shape function’. The Shepard function is then applied to construct a physically meaningful dual‐level density approximant, because of its non‐negative and range‐restricted properties. First, in terms of the original set of nodal density variables, this study develops a nonlocal nodal density approximant with enhanced smoothness by incorporating the Shepard function into the problem formulation. The density at any node can be evaluated according to the density variables located inside the influence domain of the current node. Second, in the numerical implementation, we present a point‐wise density interpolant via the Shepard function method. The density of any computational point is determined by the surrounding nodal densities within the influence domain of the concerned point. According to a set of generic design variables scattered at field nodes, an alternative solid isotropic material with penalization model is thus established through the proposed dual‐level density approximant. The Lagrangian multiplier method is included to enforce the essential boundary conditions because of the lack of the Kronecker delta function property of MLS meshless shape functions. Two benchmark numerical examples are employed to demonstrate the effectiveness of the proposed method, in particular its applicability in eliminating numerical instabilities. Copyright © 2012 John Wiley & Sons, Ltd.     

A boundary density evolutionary topology optimization of continuum structures with smooth boundaries

Based on the variable density method, this article proposes a boundary density evolutionary topology optimization method. The method uses a material interpolation model without penalization. Combined with the density grading filtering method, an optimal topology with only 0/1 cells can be obtained. Compared with the solid isotropic microstructures with penalization method (SIMP), no penalty factor is required in the material interpolation model; compared with the evolutionary structural optimization method (ESO), intermediate-density elements are allowed in the optimization process, but the concept of gradually removing the low-utilization materials near the boundary in the ESO method is retained. After the optimal result is obtained, the structural boundary element is processed by the level set of nodal strain energy, and the optimization result with smooth boundaries similar to the level set method (LSM) can be obtained. The proposed method has the superiority of the variable density method, and it also combines the advantages of the evolutionary method and the level set method, so which is named as boundary density evolution (BDE) method. The four static and one dynamic optimization examples illustrate the stability and efficiency of the proposed method.     

Piecewise constant level set method for structural topology optimization

In this paper, a piecewise constant level set (PCLS) method is implemented to solve a structural shape and topology optimization problem. In the classical level set method, the geometrical boundary of the structure under optimization is represented by the zero level set of a continuous level set function, e.g. the signed distance function. Instead, in the PCLS approach the boundary is described by discontinuities of PCLS functions. The PCLS method is related to the phase‐field methods, and the topology optimization problem is defined as a minimization problem with piecewise constant constraints, without the need of solving the Hamilton–Jacobi equation. The result is not moving the boundaries during the iterative procedure. Thus, it offers some advantages in treating geometries, eliminating the reinitialization and naturally nucleating holes when needed. In the paper, the PCLS method is implemented with the additive operator splitting numerical scheme, and several numerical and procedural issues of the implementation are discussed. Examples of 2D structural topology optimization problem of minimum compliance design are presented, illustrating the effectiveness of the proposed method. Copyright © 2008 John Wiley & Sons, Ltd.     

Phase‐field material point method for brittle fracture

The material point method for the analysis of deformable bodies is revisited and originally upgraded to simulate crack propagation in brittle media. In this setting, phase‐field modelling is introduced to resolve the crack path geometry. Following a particle in cell approach, the coupled continuum/phase‐field governing equations are defined at a set of material points and interpolated at the nodal points of an Eulerian, ie, non‐evolving, mesh. The accuracy of the simulated crack path is thus decoupled from the quality of the underlying finite element mesh and relieved from corresponding mesh‐distortion errors. A staggered incremental procedure is implemented for the solution of the discrete coupled governing equations of the phase‐field brittle fracture problem. The proposed method is verified through a series of benchmark tests while comparisons are made between the proposed scheme, the corresponding finite element implementation, and experimental results.     

Automatic mesh generation for computing symmetric local stresses by 2d-viscoplastic BEM with modified transfinite method

This paper presents a scheme of automatic mesh generation for computing the local stresses by 2D-viscoplastic boundary element method (BEM) with modified transfinite mapping method. The advanced techniques for mesh spacing and region of transfinite mapping method in BEM are studied. The applications demonstrate that the method in BEM gives high computational efficiency and precise modelling of boundaries. This reduces the time and effort required of the analyst to set up a nonlinear model, the amount of input data required is reduced greatly. The numerical examples are given in a later section.     

Topology optimization using bi-directional evolutionary structural optimization based on the element-free Galerkin method

In this article, the bi-directional evolutionary structural optimization (BESO) method based on the element-free Galerkin (EFG) method is presented for topology optimization of continuum structures. The mathematical formulation of the topology optimization is developed considering the nodal strain energy as the design variable and the minimization of compliance as the objective function. The EFG method is used to derive the shape functions using the moving least squares approximation. The essential boundary conditions are enforced by the method of Lagrange multipliers. Several topology optimization problems are presented to show the effectiveness of the proposed method. Many issues related to topology optimization of continuum structures, such as chequerboard patterns and mesh dependency, are studied in the examples.     

A level set method for shape and topology optimization of large‐displacement compliant mechanisms

A parameterization level set method is presented for structural shape and topology optimization of compliant mechanisms involving large displacements. A level set model is established mathematically as the Hamilton–Jacobi equation to capture the motion of the free boundary of a continuum structure. The structural design boundary is thus described implicitly as the zero level set of a level set scalar function of higher dimension. The radial basis function with compact support is then applied to interpolate the level set function, leading to a relaxation and separation of the temporal and spatial discretizations related to the original partial differential equation. In doing so, the more difficult shape and topology optimization problem is now fully parameterized into a relatively easier size optimization of generalized expansion coefficients. As a result, the optimization is changed into a numerical process of implementing a series of motions of the implicit level set function via an existing efficient convex programming method. With the concept of the shape derivative, the geometrical non‐linearity is included in the rigorous design sensitivity analysis to appropriately capture the large displacements of compliant mechanisms. Several numerical benchmark examples illustrate the effectiveness of the present level set method, in particular, its capability of generating new holes inside the material domain. The proposed method not only retains the favorable features of the implicit free boundary representation but also overcomes several unfavorable numerical considerations relevant to the explicit scheme, the reinitialization procedure, and the velocity extension algorithm in the conventional level set method. Copyright © 2008 John Wiley & Sons, Ltd.     

A structural optimization method based on the level set method using a new geometry‐based re‐initialization scheme

Structural optimization methods based on the level set method are a new type of structural optimization method where the outlines of target structures can be implicitly represented using the level set function, and updated by solving the so‐called Hamilton–Jacobi equation based on a Eulerian coordinate system. These new methods can allow topological alterations, such as the number of holes, during the optimization process whereas the boundaries of the target structure are clearly defined. However, the re‐initialization scheme used when updating the level set function is a critical problem when seeking to obtain appropriately updated outlines of target structures. In this paper, we propose a new structural optimization method based on the level set method using a new geometry‐based re‐initialization scheme where both the numerical analysis used when solving the equilibrium equations and the updating process of the level set function are performed using the Finite Element Method. The stiffness maximization, eigenfrequency maximization, and eigenfrequency matching problems are considered as optimization problems. Several design examples are presented to confirm the usefulness of the proposed method. Copyright © 2010 John Wiley & Sons, Ltd.