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.     

A conformal decomposition finite element method for arbitrary discontinuities on moving interfaces

Interface capturing methods using enriched finite element formulations are well suited for solving multimaterial transport problems that contain weak or strong discontinuities. The conformal decomposition FEM decomposes multimaterial elements of a non‐conforming background mesh into sub‐elements that conform to material interfaces captured using a level set method. As the interface evolves, interfacial nodes move, and background nodes may change material. The present work describes approaches for handling moving interfaces in the context of the conformal decomposition FEM for both weakly and strongly discontinuous fields. Dynamic discretization methods using extrapolation and moving mesh approaches are considered and developed with first‐order and second‐order time integration methods. The moving mesh approach is demonstrated to be a stable method that preserves both weak and strong discontinuities on a variety of one‐dimensional and two‐dimensional test problems, while achieving the expected second‐order error convergence rate in space and time. Copyright © 2014 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.     

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.     

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.     

An immersed boundary element method for level‐set based topology optimization

In this paper, we propose a new BEM for level‐set based topology optimization. In the proposed BEM, the nodal coordinates of the boundary element are replaced with the nodal level‐set function and the nodal coordinates of the Eulerian mesh that maintains the level‐set function. Because this replacement causes the nodal coordinates of the boundary element to disappear, the boundary element mesh appears to be immersed in the Eulerian mesh. Therefore, we call the proposed BEM an immersed BEM. The relationship between the nodal coordinates of the boundary element and the nodal level‐set function of the Eulerian mesh is clearly represented, and therefore, the sensitivities with respect to the nodal level‐set function are strictly derived in the immersed BEM. Furthermore, the immersed BEM completely eliminates grayscale elements that are known to cause numerical difficulties in topology optimization. By using the immersed BEM, we construct a concrete topology optimization method for solving the minimum compliance problem. We provide some numerical examples and discuss the usefulness of the constructed optimization method on the basis of the obtained results. 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 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.     

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.     

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.     

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.     

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 hierarchical method for discrete structural topology design problems with local stress and displacement constraints

In this paper, we present a hierarchical optimization method for finding feasible true 0–1 solutions to finite‐element‐based topology design problems. The topology design problems are initially modelled as non‐convex mixed 0–1 programs. The hierarchical optimization method is applied to the problem of minimizing the weight of a structure subject to displacement and local design‐dependent stress constraints. The method iteratively treats a sequence of problems of increasing size of the same type as the original problem. The problems are defined on a design mesh which is initially coarse and then successively refined as needed. At each level of design mesh refinement, a neighbourhood optimization method is used to treat the problem considered. The non‐convex topology design problems are equivalently reformulated as convex all‐quadratic mixed 0–1 programs. This reformulation enables the use of methods from global optimization, which have only recently become available, for solving the problems in the sequence. Numerical examples of topology design problems of continuum structures with local stress and displacement constraints are presented. Copyright © 2006 John Wiley & Sons, Ltd.     

An enhanced genetic algorithm for structural topology optimization

Genetic algorithms (GAs) have become a popular optimization tool for many areas of research and topology optimization an effective design tool for obtaining efficient and lighter structures. In this paper, a versatile, robust and enhanced GA is proposed for structural topology optimization by using problem‐specific knowledge. The original discrete black‐and‐white (0–1) problem is directly solved by using a bit‐array representation method. To address the related pronounced connectivity issue effectively, the four‐neighbourhood connectivity is used to suppress the occurrence of checkerboard patterns. A simpler version of the perimeter control approach is developed to obtain a well‐posed problem and the total number of hinges of each individual is explicitly penalized to achieve a hinge‐free design. To handle the problem of representation degeneracy effectively, a recessive gene technique is applied to viable topologies while unusable topologies are penalized in a hierarchical manner. An efficient FEM‐based function evaluation method is developed to reduce the computational cost. A dynamic penalty method is presented for the GA to convert the constrained optimization problem into an unconstrained problem without the possible degeneracy. With all these enhancements and appropriate choice of the GA operators, the present GA can achieve significant improvements in evolving into near‐optimum solutions and viable topologies with checkerboard free, mesh independent and hinge‐free characteristics. Numerical results show that the present GA can be more efficient and robust than the conventional GAs in solving the structural topology optimization problems of minimum compliance design, minimum weight design and optimal compliant mechanisms design. It is suggested that the present enhanced GA using problem‐specific knowledge can be a powerful global search tool for structural topology optimization. Copyright © 2005 John Wiley & Sons, Ltd.     

An adaptive level set method based on two‐level uniform meshes and its application to dislocation dynamics

In this paper, we present an adaptive level set method for motion of high codimensional objects (e.g., curves in three dimensions). This method uses only two (or a few fixed) levels of meshes. A uniform coarse mesh is defined over the whole computational domain. Any coarse mesh cell that contains the moving object is further divided into a uniform fine mesh. The coarse‐to‐fine ratios in the mesh refinement can be adjusted to achieve optimal efficiency. Refinement and coarsening (removing the fine mesh within a coarse grid cell) are performed dynamically during the evolution. In this adaptive method, the computation is localized mostly near the moving objects; thus, the computational cost is significantly reduced compared with the uniform mesh over the whole domain with the same resolution. In this method, the level set equations can be solved on these uniform meshes of different levels directly using standard high‐order numerical methods. This method is examined by numerical examples of moving curves and applications to dislocation dynamics simulations. This two‐level adaptive method also provides a basis for using locally varying time stepping to further reduce the computational cost. Copyright © 2013 John Wiley & Sons, Ltd.     

A sub‒domain smoothed Galerkin method for solid mechanics problems

A sub?domain smoothed Galerkin method is proposed to integrate the advantages of mesh?free Galerkin method and FEM. Arbitrarily shaped sub?domains are predefined in problems domain with mesh?free nodes. In each sub?domain, based on mesh?free Galerkin weak formulation, the local discrete equation can be obtained by using the moving Kriging interpolation, which is similar to the discretization of the high?order finite elements. Strain smoothing technique is subsequently applied to the nodal integration of sub?domain by dividing the sub?domain into several smoothing cells. Moreover, condensation of DOF can also be introduced into the local discrete equations to improve the computational efficiency. The global governing equations of present method are obtained on the basis of the scheme of FEM by assembling all local discrete equations of the sub?domains. The mesh?free properties of Galerkin method are retained in each sub?domain. Several 2D elastic problems have been solved on the basis of this newly proposed method to validate its computational performance. These numerical examples proved that the newly proposed sub?domain smoothed Galerkin method is a robust technique to solve solid mechanics problems based on its characteristics of high computational efficiency, good accuracy, and convergence. Copyright © 2014 John Wiley & Sons, Ltd.     

Dispersion and pollution of the FEM solution for the Helmholtz equation in one,two and three dimensions

For high wave numbers, the Helmholtz equation suffers the so‐called ‘pollution effect’. This effect is directly related to the dispersion. A method to measure the dispersion on any numerical method related to the classical Galerkin FEM is presented. This method does not require to compute the numerical solution of the problem and is extremely fast. Numerical results on the classical Galerkin FEM (p‐method) is compared to modified methods presented in the literature. A study of the influence of the topology triangles is also carried out. The efficiency of the different methods is compared. The numerical results in two of the mesh and for square elements show that the high order elements control the dispersion well. The most effective modified method is the QSFEM [1,2] but it is also very complicated in the general setting. The residual‐free bubble [3,4] is effective in one dimension but not in higher dimensions. The least‐square method [1,5] approach lowers the dispersion but relatively little. The results for triangular meshes show that the best topology is the ‘criss‐cross’ pattern. Copyright © 1999 John Wiley & Sons, Ltd.     

Arbitrary placement of local meshes in a global mesh by the interface‐element method (IEM)

A new method is proposed to place local meshes in a global mesh with the aid of the interface‐element method (IEM). The interface‐elements use moving least‐square (MLS)‐based shape functions to join partitioned finite‐element domains with non‐matching interfaces. The supports of nodes are defined to satisfy the continuity condition on the interfaces by introducing pseudonodes on the boundaries of interface regions. Particularly, the weight functions of nodes on the boundaries of interface regions span only neighbouring nodes, ensuring that the resulting shape functions are identical to those of adjoining finite‐elements. The completeness of the shape functions of the interface‐elements up to the order of basis provides a reasonable transfer of strain fields through the non‐matching interfaces between partitioned domains. Taking these great advantages of the IEM, local meshes can be easily inserted at arbitrary places in a global mesh. Several numerical examples show the effectiveness of this technique for modelling of local regions in a global domain. Copyright © 2003 John Wiley & Sons, Ltd.     

A finite element method with mesh‐separation‐based approximation technique and its application in modeling crack propagation with adaptive mesh refinement

This paper presents a FEM with mesh‐separation‐based approximation technique that separates a standard element into three geometrically independent elements. A dual mapping scheme is introduced to couple them seamlessly and to derive the element approximation. The novel technique makes it very easy for mesh generation of problems with complex or solution‐dependent, varying geometry. It offers a flexible way to construct displacement approximations and provides a unified framework for the FEM to enjoy some of the key advantages of the Hansbo and Hansbo method, the meshfree methods, the semi‐analytical FEMs, and the smoothed FEM. For problems with evolving discontinuities, the method enables the devising of an efficient crack‐tip adaptive mesh refinement strategy to improve the accuracy of crack‐tip fields. Both the discontinuities due to intra‐element cracking and the incompatibility due to hanging nodes resulted from the element refinement can be treated at the elemental level. The effectiveness and robustness of the present method are benchmarked with several numerical examples. The numerical results also demonstrate that a high precision integral scheme is critical to pass the crack patch test, and it is essential to apply local adaptive mesh refinement for low fracture energy problems. Copyright © 2014 John Wiley & Sons, Ltd.