Topological Shape Optimization Method Research Articles

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

SummaryThe 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.

Topology optimization for concurrent design of structures with multi-patch microstructures by level sets

This paper focuses on the novel concurrent design for cellular structures consisting of multiple patches of material microstructures using a level set-based topological shape optimization method. The macro structure is featured with the configuration of a cluster of non-uniformly distributed patches, while each patch hosts a number of identical material microstructures. At macro scale, a discrete element density based approach is presented to generate an overall structural layout involving different groups of discrete element densities. At micro scale, each macro element is regarded as an individual microstructure with a discrete intermediate density. Hence, all the macro elements with the same discrete densities (volume fractions) are represented by a unique microstructure. The representative microstructures corresponding to different density groups are topologically optimized by incorporating the numerical homogenization approach into a parametric level set method. The multiscale concurrent designs are integrated into a uniform optimization procedure, so as to optimize both topologies for the macrostructure and its microstructures, as well as locations of the microstructures in the design space. Numerical examples demonstrate that the proposed method can substantially improve the structural performance with an affordable computation and manufacturing cost.

Isogeometric topological shape optimization using dual evolution with boundary integral equation and level sets

An isogeometric topological shape optimization method is developed, using a dual evolution of NURBS curves and level sets; the NURBS curves feature the exact representation of geometry and the level sets help to detect and guide the topological variation of NURBS curves. The implicit geometry by the level sets is transformed into the parametric NURBS curves by minimizing the difference of velocity fields in both representations. A gradient-based optimization problem is formulated, based on the evolution of the NURBS curves. The control points of NURBS curves are taken as design variables. The necessary response and design sensitivity are computed by an isogeometric boundary integral equation method (BIEM) using the NURBS curves. The design sensitivity is obtained on fixed grids and utilized as the velocity to update the Hamilton–Jacobi equation for the level sets. To obtain the whole velocity field on the fixed grids, an interpolation and velocity extension scheme are employed. The developed method provides accurate response and enhanced sensitivity using isogeometric BIEM. Also, additional post-processing is not required to communicate with CAD systems since the optimal design is represented as NURBS curves. Numerical examples demonstrate the accuracy of design sensitivity on fixed grids and the feasibility of shape and topological optimization.

Topological design for mechanical metamaterials using a multiphase level set method

Metamaterials usually refer to artificially engineered composites with unusual properties that cannot be easily found in nature. This paper will develop a topological shape optimization method for design of mechanical metamaterials of thermoelastic micro-structured composites, which integrates numerical homogenization method with a multi-phase level set method (MPLSM). The homogenization method is applied to evaluate the effective macroscopic properties of a periodic microstructure, while the MPLSM will be utilized to implement shape and topology evolutions of the microstructure. A multi-phase level set representation model is established to describe the boundaries of the multi-material microstructure using a combination of all level set functions, without overlaps and empties. The Hamilton-Jacobi partial differential equation-based topological shape optimization problem will be transformed to a generalized size optimization problem. Typical numerical examples are used to demonstrate the effectiveness of the proposed method for designing metamaterials with expected and extreme thermal expansion coefficients.

Topological shape optimization of microstructural metamaterials using a level set method

Metamaterials usually refer to artificial composite materials consisting of an array of periodically arranged microstructures, engineered to provide unusual material properties that may not be easily found in nature. This paper proposes a new topological shape optimization method for systematic computational design of a type of mechanical metamaterials with negative Poisson’s ratios (auxetic materials), which integrates the numerical homogenization approach into a powerful parametric level set method (PLSM). The homogenization method is used to obtain the effective properties of the periodic microstructure, while the PLSM is applied to achieve shape evolutions and topological changes of the microstructure, until the desired material properties are achieved. The key concept of the PLSM is the interpolation of the implicit level set surface by using a given set of compactly supported radial basis functions (CSRBF), which are positioned at a number of given and fixed knots inside the design domain. Several typical numerical examples are used to demonstrate the favorable characteristics of the proposed method in the design of micro-structured metamaterials.

Level Set Based Shape Optimization of Linear Structures using Topological Derivatives

Abstract Using a level set method and topological derivatives, a topological shape optimization method that is independent of an initial design is developed for linearly elastic structures. In the level set method, the initial domain is kept fixed and its boundary is represented by an implicit moving boundary embedded in the level set function, which facilitates to handle complicated topological shape changes. The “Hamilton-Jacobi(H-J)” equation and computationally robust numerical technique of “up-wind scheme” lead the initial implicit boundary to an optimal one according to the normal velocity field while minimizing the objective function of compliance and satisfying the constraint of allowable volume. Based on the asymptotic regularization concept, the topological derivative is considered as the limit of shape derivative as the radius of hole approaches to zero. The required velocity field to update the H-J equation is determined from the descent direction of Lagrangian derived from optimality conditions. It turns out that the initial holes are not required to get the optimal result since the developed method can create holes whenever and wherever necessary using indicators obtained from the topological derivatives. It is demonstrated that the proper choice of control parameters for nucleation is crucial for efficient optimization process.

Level Set Based Topological Shape Optimization Combined with Meshfree Method

Using the level set and the meshfree methods, we develop a topological shape optimization method applied to linear elasticity problems. Design gradients are computed using an efficient adjoint design sensitivity analysis(DSA) method. The boundaries are represented by an implicit moving boundary(IMB) embedded in the level set function obtainable from the “Hamilton-Jacobi type” equation with the “Up-wind scheme.” Then, using the implicit function, explicit boundaries are generated to obtain the response and sensitivityof the structures. Global nodal shape function derived on a basis of the reproducing kernel(RK) method is employed to discretize the displacement field in the governing continuum equation. Thus, the material points can be located everywhere in the continuum domain, which enables to generate the explicit boundaries and leads to a precise design result. The developed method defines a Lagrangian functional for the constrained optimization. It minimizes the compliance, satisfying the constraint of allowable volume through the variations of boundary. During the optimization, the velocity to integrate the Hamilton-Jacobi equation is obtained from the optimality condition for the Lagrangian functional. Compared with the conventional shape optimization method, the developed one can easily represent the topological shape variations.

헤비사이드 강화를 이용한 구조물의 아이소-지오메트릭 위상 최적설계

An isogeometric topological shape optimization method is developed using the level sets and Heaviside enrichments. In the level set method, the initial domain is kept fixed and its boundary is represented by an implicit moving boundary embedded in the level set functions, which facilitates to handle complicated topological shape changes. The Heaviside enrichment improves the isogeometric analysis by adding some enrichment functions to model the internal boundaries. The proposed topological shape optimization method has several benefits: exact geometric models can be obtained using the isogeometric approach and the limitation of tensor-product patches can be overcome using the Heaviside enrichments to represent the internal voids. Even in a single patch, discontinuous displacement fields as well as smooth stress field can be obtained. Since the level sets offer the implicit moving boundary inside the domain, it is easy to represent the topological shape variations in the isogeometric analysis using Heaviside enrichments.

위상 민감도를 이용한 초탄성 비선형 구조의 레벨셋 기반 위상 및 형상 최적설계

A level set based topological shape optimization method for nonlinear structure considering hyper-elastic problems is developed. To relieve significant convergence difficulty in topology optimization of nonlinear structure due to inaccurate tangent stiffness which comes from material penalization of whole domain, explicit boundary for exact tangent stiffness is used by taking advantage of level set function for arbitrary boundary shape. For given arbitrary boundary which is represented by level set function, a Delaunay triangulation scheme is used for current structure discretization instead of using implicit fixed grid. The required velocity field in the actual domain to update the level set equation is determined from the descent direction of Lagrangian derived from optimality conditions. The velocity field outside the actual domain is determined through a velocity extension scheme based on the method suggested by Adalsteinsson and Sethian(1999). The topological derivatives are incorporated into the level set based framework to enable to create holes whenever and wherever necessary during the optimization.

Level Set–Based Topological Shape Optimization of Heat Conduction Problems Considering Design-Dependent Convection Boundary

A level set–based topological shape optimization method considering design-dependent convection boundaries is developed for steady-state heat conduction problems. We embed the level set function obtained from a Hamilton-Jacobi type of equation into a fixed initial domain to implicitly represent thermal boundaries. The effects of the implicit convection boundary obtained from topological shape variations are represented by numerical Dirac delta and Heaviside functions. The method minimizes the thermal compliance of systems by varying the implicit boundary, satisfying the constraint of allowable material volume. During design optimization, the boundary velocity to integrate the Hamilton-Jacobi equation is derived from an optimality condition.

Level Set-Based Topological Shape Optimization of Nonlinear Heat Conduction Problems Using Topological Derivatives#

A level set-based topological shape-optimization method is developed to relieve the well-known convergence difficulty in nonlinear heat-conduction problems. While minimizing the objective function of instantaneous thermal compliance and satisfying the constraint of allowable volume, the solution of the Hamilton–Jacobi equation leads the initial implicit boundary to an optimal one according to the normal velocity determined from the descent direction of the Lagrangian. Topological derivatives are incorporated into the level set-based framework to improve convergence of the optimization process as well as to avoid the local minimum resulting from the intrinsic nature of the shape-design approach.

Level Set-based Topological Shape Optimization of Nonlinear Heat Conduction Problems

A level set-based topological shape optimization method is developed for nonlinear heat conduction problems. While minimizing the objective function of instantaneous thermal compliance and satisfying the constraint of allowable volume, solution of the Hamilton-Jacobi equation leads the initial boundary to an optimal one according to the normal velocity field determined from the descent direction of the Lagrangian. To overcome the convergence difficulty in nonlinear problems resulting from introduction of an approximate boundary, an actual boundary is identified by tracking the level set functions and remeshing using Delaunay triangulation. The velocity field outside the actual domain is determined through a velocity extension scheme.

Level set based topological shape optimization of geometrically nonlinear structures using unstructured mesh

Using hyperelastic materials and unstructured mesh, a level set based topological shape optimization method is developed for geometrically nonlinear structures in total Lagrangian framework. In the level set method, the initial domain is kept fixed and its boundary is represented by an implicit moving boundary embedded in a level set function, which facilitates to handle complicated topological shape changes and eventually leads the initial implicit boundary to an optimal one according to the normal velocity field while both minimizing the objective function of instantaneous structural compliance and satisfying an allowable material volume constraint. In existing level set based methods, an initial reference domain or an ersatz material is employed for the penalization of whole domain to represent the current domain. However, these approaches end up with a convergence difficulty in nonlinear response analysis due to the inaccurate tangent stiffness. To overcome this difficulty, taking advantage of the obtained level set function, the current structural boundary is actually represented using a Delaunay triangulation scheme and a hyperelastic material law is employed to handle the large strain problem. The required velocity field in the actual domain to update the level set equation is determined from the descent direction of Lagrangian derived from optimality conditions. The velocity field outside the actual domain is determined through a velocity extension scheme based on a fast marching method. Since homogeneous material property and actual boundary are utilized, the convergence difficulty is significantly relieved.

Topological Shape Optimization of Heat Conduction Problems using Level Set Approach

ABSTRACT A topological shape optimization method for heat conduction problems is developed using a level set method. The level set function obtained from the “Hamilton-Jacobi type” equation is embedded into a fixed initial domain to implicitly represent thermal boundaries and obtain the finite-element response and adjoint sensitivity. The developed method minimizes the thermal compliance, satisfying the constraint of allowable volume by varying the implicit boundary. During optimization, the boundary velocity to integrate the Hamilton-Jacobi equation is obtained from the optimality condition. The newly developed method shows no numerical instability and makes it easy to represent topological shape variations.

Topological shape optimization of geometrically nonlinear structures using level set method

Using the level set method, a topological shape optimization method is developed for geometrically nonlinear structures in total Lagrangian formulation. The structural boundaries are implicitly represented by the level set function, obtainable from “Hamilton–Jacobi type” equation with “up-wind scheme,” embedded into a fixed initial domain. The method minimizes the compliance through the variations of implicit boundary, satisfying an allowable volume requirement. The required velocity field to solve the Hamilton–Jacobi equation is determined by the descent direction of Lagrangian derived from an optimality condition. Since the homogeneous material property and implicit boundary are utilized, the convergence difficulty is significantly relieved.

Topological shape optimization of power flow problems at high frequencies using level set approach

Using a level set method we develop a topological shape optimization method applied to power flow problems in steady state. Necessary design gradients are computed using an efficient adjoint sensitivity analysis method. The boundaries are implicitly represented by the level set function obtainable from the “Hamilton–Jacobi type” equation with the “Up-wind scheme.” The implicit function is embedded into a fixed initial domain to obtain the finite element response and sensitivity. The developed method defines a Lagrangian function for the constrained optimization. It minimizes a generalized compliance, satisfying the constraint of allowable volume through the variations of implicit boundary. During the optimization, the boundary velocity to integrate the Hamilton–Jacobi equation is obtained from the optimality condition for the Lagrangian function. Compared with the established topology optimization method, the developed one has no numerical instability such as checkerboard problems and easy representation of topological shape variations.