Structural shape and topology optimization of cast parts using level set method

This paper presents a level set‐based shape and topology optimization method for conceptual design of cast parts. In order to be successfully manufactured by the casting process, the geometry of cast parts should satisfy certain moldability conditions, which poses additional constraints in the shape and topology optimization of cast parts. Instead of using the originally point‐wise constraint statement, we propose a casting constraint in the form of domain integration over a narrowband near the material boundaries. This constraint is expressed in terms of the gradient of the level set function defining the structural shape and topology. Its explicit and analytical form facilitates the sensitivity analysis and numerical implementation. As compared with the standard implementation of the level set method based on the steepest descent algorithm, the proposed method uses velocity field design variables and combines the level set method with the gradient‐based mathematical programming algorithm on the basis of the derived sensitivity scheme of the objective function and the constraints. This approach is able to simultaneously account for the casting constraint and the conventional material volume constraint in a convenient way. In this method, the optimization process can be started from an arbitrary initial design, without the need for an initial design satisfying the cast constraint. Numerical examples in both 2D and 3D design domain are given to demonstrate the validity and effectiveness of the proposed method. Copyright © 2017 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.     

Radial basis functions and level set method for structural topology optimization

Level set methods have become an attractive design tool in shape and topology optimization for obtaining lighter and more efficient structures. In this paper, the popular radial basis functions (RBFs) in scattered data fitting and function approximation are incorporated into the conventional level set methods to construct a more efficient approach for structural topology optimization. RBF implicit modelling with multiquadric (MQ) splines is developed to define the implicit level set function with a high level of accuracy and smoothness. A RBF–level set optimization method is proposed to transform the Hamilton–Jacobi partial differential equation (PDE) into a system of ordinary differential equations (ODEs) over the entire design domain by using a collocation formulation of the method of lines. With the mathematical convenience, the original time dependent initial value problem is changed to an interpolation problem for the initial values of the generalized expansion coefficients. A physically meaningful and efficient extension velocity method is presented to avoid possible problems without reinitialization in the level set methods. The proposed method is implemented in the framework of minimum compliance design that has been extensively studied in topology optimization and its efficiency and accuracy over the conventional level set methods are highlighted. Numerical examples show the success of the present RBF–level set method in the accuracy, convergence speed and insensitivity to initial designs in topology optimization of two‐dimensional (2D) structures. It is suggested that the introduction of the radial basis functions to the level set methods can be promising in structural topology optimization. Copyright © 2005 John Wiley & Sons, Ltd.     

Velocity field level-set method for topological shape optimization using freely distributed design variables

The velocity field level-set topological shape optimization method combines the implicit representation in the standard level-set method and the capabilities of general mathematical programming algorithms in handling multiple constraints and additional design variables. The key concept is to construct the normal velocity field using basis functions and the velocity design variables at specified points (referred to as velocity knots) in the entire design domain. In this study, the velocity design variables are decoupled from the level-set grid points. Making use of this property, we can adaptively change the arrangement of the velocity knots as the structural boundary evolves. This provides more design freedom in the optimization and allows for a significant reduction in the number of design variables. Several numerical examples in two- and three-dimensional design domains are presented to demonstrate the robustness and efficiency of the proposed method. We also show that changing the number of velocity knots may implicitly exert certain control on topological complexity and length scale.     

Parametric shape and topology optimization: A new level set approach based on cardinal basis functions

The parametric level set approach is an extension of the conventional level set methods for topology optimization. By parameterizing the level set function, level set methods can be directly coupled with mathematical programming to achieve better numerical robustness and computational efficiency. Moreover, the parametric level set scheme can not only inherit the primary advantages of the conventional level set methods, such as clear boundary representation and the flexibility in handling topological changes, but also alleviate some undesired features from the conventional level set methods, such as the need for reinitialization. However, in the existing radial basis function–based parametric level set method, it is difficult to identify the range of the design variables. Besides, the parametric level set evolution often struggles with large fluctuations during the optimization process. Those issues cause difficulties both in numerical stability and in material property mapping. In this paper, a cardinal basis function is constructed based on the radial basis function partition of unity collocation method to parameterize the level set function. The benefit of using cardinal basis function is that the range of the design variables can now be clearly specified as the value of the level set function. A distance regularization energy functional is also introduced, aiming to maintain the desired signed distance property during the level set evolution. With this desired feature, the level set evolution is stabilized against large fluctuations. In addition, the material properties mapped from the level set function to the finite element model can be more accurate.     

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.     

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.     

Dual methods for discrete structural optimization problems

The purpose of this paper is to present a mathematical programming method developed to solve structural optimization problems involving discrete variables. We work in the following context: the structural responses are computed by the finite elements method and convex and separable approximation schemes are used to generate a sequence of explicit approximate subproblems.Each of them is solved in the dual space with a subgradient‐based algorithm (or with a variant of it) specially developed to maximize the not everywhere differentiable dual function. To show that the application field is large, the presented applications are issued from different domains of structural design, such as sizing of thin‐walled structures, geometrical configuration of trusses, topology optimization of membrane or 3‐D structures and welding points numbering in car bodies. The main drawback of using the dual approach is that the obtained solution is generally not the global optimum. This is linked to the presence of a duality gap, due to the non‐convexity of the primal discrete subproblems. Fortunately, this gap can be quantified: a maximum bound on its value can be computed. Moreover, it turns out that the duality gap is decreasing for higher number of variables; the maximum bound on the duality gap is generally negligible in the treated applications. The developed algorithms are very efficient for 2‐D and 3‐D topology optimization, where applications involving thousands of binary design variables are solved in a very short time. Copyright © 2000 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.     

Level set topology optimization of structural problems with interface cohesion

This paper presents a finite element topology optimization framework for the design of two‐phase structural systems considering contact and cohesion phenomena along the interface. The geometry of the material interface is described by an explicit level set method, and the structural response is predicted by the extended finite element method. In this work, the interface condition is described by a bilinear cohesive zone model on the basis of the traction‐separation constitutive relation. The non‐penetration condition in the presence of compressive interface forces is enforced by a stabilized Lagrange multiplier method. The mechanical model assumes a linear elastic isotropic material, infinitesimal strain theory, and a quasi‐static response. The optimization problem is solved by a nonlinear programming method, and the design sensitivities are computed by the adjoint method. The performance of the presented method is evaluated by 2D and 3D numerical examples. The results obtained from topology optimization reveal distinct design characteristics for the various interface phenomena considered. In addition, 3D examples demonstrate optimal geometries that cannot be fully captured by reduced dimensionality. The optimization framework presented is limited to two‐phase structural systems where the material interface is coincident in the undeformed configuration, and to structural responses that remain valid considering small strain kinematics. Copyright © 2017 John Wiley & Sons, Ltd.     

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.     

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.     

A nodal variable method of structural topology optimization based on Shepard interpolant

A method for topology optimization of continuum structures based on nodal density variables and density field mapping technique is investigated. The original discrete‐valued topology optimization problem is stated as an optimization problem with continuous design variables by introducing a material density field into the design domain. With the use of the Shepard family of interpolants, this density field is mapped onto the design space defined by a finite number of nodal density variables. The employed interpolation scheme has an explicit form and satisfies range‐restricted properties that makes it applicable for physically meaningful density interpolation. Its ability to resolve more complex spatial distribution of the material density within an individual element, as compared with the conventional elementwise design variable approach, actually provides certain regularization to the topology optimization problem. Numerical examples demonstrate the validity and applicability of the proposed formulation and numerical techniques. Copyright © 2011 John Wiley & Sons, Ltd.     

Level set‐based topology optimization targeting dielectric resonator‐based composite right‐ and left‐handed transmission lines

In the last decade, metamaterials have been gaining attention and have been investigated because of their unique characteristics, which conventional materials do not have, such as negative refraction indexes. However, it is sometimes difficult to design metamaterials on the basis of experience and theoretical considerations because the relationship between their electromagnetic characteristics and structure is often vague. A mathematical structural design methodology targeting metamaterials may therefore be useful for expanding the engineering applications of metamaterials in industry. In this paper, a new level set‐based topology optimization method is proposed for designing composite right‐ and left‐handed transmission lines, each of which consists of a waveguide and periodically located dielectric resonators. Such transmission lines function as a fundamental metamaterial. In the proposed method, the shape and topology of the dielectric resonators are represented by the level set function, and topology optimization problems are formulated on the basis of the level set‐based representation. Copyright © 2011 John Wiley & Sons, Ltd.     

Optimization of Construction of Tire Reinforcement by Genetic Algorithm

A new tire design procedure capable of determining the optimum tire construction was developed by combining a finite element method approach with mathematical programming and a genetic algorithm (GA). Both procedures successfully generated optimized belt structures. The design variables in the mathematical programming were belt angle and belt width. Using the merits of a GA which enabled the use of discrete variables, the design variables in the GA were not only the topology of the belt and belt angle but also the belt material. Furthermore, a discrete objective function such as the number of parts could be optimized in the GA. The optimized structure obtained by the GA was verified to increase the cornering stiffness more than 15 percent as compared with the control structure in an indoor drum test.     

Topology optimization with multiple materials via moving morphable component (MMC) method

With the fast development of additive manufacturing technology, topology optimization involving multiple materials has received ever increasing attention. Traditionally, this kind of optimization problem is solved within the implicit solution framework by using the Solid Isotropic Material with Penalization or level set method. This treatment, however, will inevitably lead to a large number of design variables especially when many types of materials are involved and 3‐dimensional (3D) problems are considered. This is because for each type of material, a corresponding density field/level function defined on the entire design domain must be introduced to describe its distribution. In the present paper, a novel approach for topology optimization with multiple materials is established based on the Moving Morphable Component framework. With use of this approach, topology optimization problems with multiple materials can be solved with much less numbers of design variables and degrees of freedom. Numerical examples provided demonstrate the effectiveness of the proposed approach.     

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.     

Topology optimization by a time‐dependent diffusion equation

Most topology optimization problems are formulated as constrained optimization problems; thus, mathematical programming has been the mainstream. On the other hand, solving topology optimization problems using time evolution equations, seen in the level set‐based and the phase field‐based methods, is yet another approach. One issue is the treatment of multiple constraints, which is difficult to incorporate within time evolution equations. Another issue is the extra re‐initialization steps that interrupt the time integration from time to time. This paper proposes a way to describe, using a Heaviside projection‐based representation, a time‐dependent diffusion equation that addresses these two issues. The constraints are treated using a modified augmented Lagrangian approach in which the Lagrange multipliers are updated by simple ordinary differential equations. The proposed method is easy to implement using a high‐level finite element code. Also, it is very practical in the sense that one can fully utilize the existing framework of the code: GUI, parallelized solvers, animations, data imports/exports, and so on. The effectiveness of the proposed method is demonstrated through numerical examples in both the planar and spatial cases. Copyright © 2012 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.