Seung-Hyun Ha

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.

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

Abstract 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.

Isogeometric Shape Design Optimization of Geometrically Nonlinear Structures

Using the isogeometric approach, the variational formulation of both response and sensitivity analyses for geometrically nonlinear structures is derived using the total Lagrangian formulation. The geometric properties of design are embedded in the nonuniform rational B-splines basis functions and control points whose perturbation naturally results in shape changes. Thus, exact geometric models can be utilized in both response and sensitivity analyses. The normal vector and curvature are continuous in the whole design space so that enhanced shape sensitivity can be expected. Refinements and design changes during the shape optimization are easily implemented within the isogeometric framework, which maintains exact geometry without subsequent communication with computer-aided design description. Through numerical examples, the developed isogeometric sensitivity is verified to demonstrate excellent agreement with finite difference sensitivity. Also, the proposed method works very well in various shape optimization problems.

Level Set Based Topological Shape Optimization of Hyper-elastic Nonlinear Structures using Topological Derivatives

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.

Design Sensitivity Analysis of Coupled MD-Continuum Systems Using Bridging Scale Approach

We present a design sensitivity analysis(DSA) method for multiscale problems based on bridging scale decomposition. In this paper, we utilize a bridging scale method for the coupled system analysis. Since the analysis of full MD systems requires huge amount of computational costs, a coupled system of MD-level and continuum-level simulation is usually preferred. The information exchange between the MD and continuum levels is taken place at the MD-continuum boundary. In the bridging scale method, a generalized Langevin equation(GLE) is introduced for the reduced MD system and the GLE force using a time history kernel is applied at the boundary atoms in the MD system. Therefore, we can separately analyze the MD and continuum level simulations, which can accelerate the computing process. Once the simulation of coupled problems is successful, the need for the DSA is naturally arising for the optimization of macro-scale design, where the macro scale performance of the system is maximized considering the micro scale effects. The finite difference sensitivity is impractical for the gradient based optimization of large scale problems due to the restriction of computing costs but the analytical sensitivity for the coupled system is always accurate. In this study, we derive the analytical design sensitivity to verify the accuracy and applicability to the design optimization of the coupled system.

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.

Shape Design Optimization of Crack Propagation Problems Using Meshfree Methods

This paper presents a continuum-based shape design sensitivity analysis(DSA) method for crack propagation problems using a reproducing kernel method(RKM), which facilitates the remeshing problem required for finite element analysis(FEA) and provides the higher order shape functions by increasing the continuity of the kernel functions. A linear elasticity is considered to obtain the required stress field around the crack tip for the evaluation of J-integral. The sensitivity of displacement field and stress intensity factor(SIF) with respect to shape design variables are derived using a material derivative approach. For efficient computation of design sensitivity, an adjoint variable method is employed tather than the direct differentiation method. Through numerical examples, The mesh-free and the DSA methods show excellent agreement with finite difference results. The DSA results are further extended to a shape optimization of crack propagation problems to control the propagation path.

Isogeometric Shape Design Optimization of Power Flow Problems at High Frequencies

Abstract Using an isogeometric approach, a continuum-based shape design optimization method is developed for steady state power flow problems at high frequencies. In case the isogeometric method is employed to the shape design optimization, the NURBS basis functions used in CAD geometric modeling are directly utilized to embed the exact geometry into the computational framework so that the design parameterization for shape optimization is much easier than that in the finite element method and consequently provides the enhanced smoothness of design perturbations. Thus, exact geometric models can be used in both the response and the shape sensitivity analyses, where normal vector and curvature are continuous over the whole design space so that enhanced shape sensitivity can be expected. Through numerical examples, the developed isogeometric sensitivity is compared with finite difference one to provide excellent agreement. Also, it turns out that the proposed method works very well in the shape optimization problems.

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.