Min-Geun Kim

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.

Adjoint design sensitivity analysis of reduced atomic systems using generalized Langevin equation for lattice structures

An efficient adjoint design sensitivity analysis method is developed for reduced atomic systems. A reduced atomic system and the adjoint system are constructed in a locally confined region, utilizing generalized Langevin equation (GLE) for periodic lattice structures. Due to the translational symmetry of lattice structures, the size of time history kernel function that accounts for the boundary effects of the reduced atomic systems could be reduced to a single atoms degrees of freedom. For the problems of highly nonlinear design variables, the finite difference method is impractical for its inefficiency and inaccuracy. However, the adjoint method is very efficient regardless of the number of design variables since one additional time integration is required for the adjoint GLE. Through numerical examples, the derived adjoint sensitivity turns out to be accurate and efficient through the comparison with finite difference sensitivity.

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.

Shape-design optimization of hull structures considering thermal deformation

We developed a shape-design optimization method for the thermo-elastoplasticity problems that are applicable to the welding or thermal deformation of hull structures. The point is to determine the shape-design parameters such that the deformed shape after welding fits very well to a desired design. The geometric parameters of curved surfaces are selected as the design parameters. The shell finite elements, forward finite difference sensitivity, modified method of feasible direction algorithm and a programming language ANSYS Parametric Design Language in the established code ANSYS are employed in the shape optimization. The objective function is the weighted summation of differences between the deformed and the target geometries. The proposed method is effective even though new design variables are added to the design space during the optimization process since the multiple steps of design optimization are used during the whole optimization process. To obtain the better optimal design, the weights are determined for the next design optimization, based on the previous optimal results. Numerical examples demonstrate that the localized severe deviations from the target design are effectively prevented in the optimal design.

Level Set based Topological Shape Optimization of Phononic Crystals

A topology optimization method for phononic crystals is developed for the design of sound barriers, using the level set approach. Given a frequency and an incident wave to the phononic crystals, an optimal shape of periodic inclusions is found by minimizing the norm of transmittance. In a sound field including scattering bodies, an acoustic wave can be refracted on the obstacle boundaries, which enables to control acoustic performance by taking the shape of inclusions as the design variables. In this research, we consider a layered structure which is composed of inclusions arranged periodically in horizontal direction while finite inclusions are distributed in vertical direction. Due to the periodicity of inclusions, a unit cell can be considered to analyze the wave propagation together with proper boundary conditions which are imposed on the left and right edges of the unit cell using the Bloch theorem. The boundary conditions for the lower and the upper boundaries of unit cell are described by impedance matrices, which represent the transmission of waves between the layered structure and the semi-infinite external media. A level set method is employed to describe the topology and the shape of inclusions. 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. Through several numerical examples, the applicability of the proposed method is demonstrated.

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.

Adjoint Design Sensitivity Analysis of Fracture Mechanics Using Molecular-Continuum Multiscale Approach

We have developed a multiscale design sensitivity analysis method for transient dynamics using a bridging scale method by a projection operator for scale decomposition. Employing a mass-weighted projection operator, we can fully decouple the equations of motion into fine and coarse scales using the orthogonal property of complimentary projector to the mass matrix. Thus, independent solvers in response analyses can be utilized for the fine scale analysis of molecular dynamic (MD) and the coarse scale analysis of finite element analysis. To reduce the size of problems and to improve the computational efficiency, a generalized Langevin equation is used for a localized MD analysis. Through demonstrative numerical examples, it turns out that the derived sensitivity analysis method is accurate and efficient compared with finite difference sensitivity.Copyright

Shape Design Optimization of Ship Structures Considering Thermal Deformation and Target Shape

In this paper, we develop a shape design optimization method for thermo-elasto- plasticity problems that is applicable to the welding or thermal deformation problems of ship structures. Shell elements and a programming language APDL in a commercial finite element analysis code, ANSYS, are employed in the shape optimiz ation. The point of developed method is to determine the design parameters such that the deformed shape after welding fits very well to a desired design. The geometric parameters of surfaces are selected as the design parameters. The modified method of feasible direction (MMFD) and finite difference sensitivity are used for the optimization alg orithm. Two numerical examples demonstrate that the developed shape design method is applicable to existing hull structures and effective for the structural design of ships. ※Keywords: Thermo-elastic-plastic analysis (열탄소성 해석), Welding deformation (용접 변형), Shape design optimization (형상 최적설계), Finite difference sensitiv ity (유한차분 민감도), Method of feasible direction (유용 방향법)

Level Set Based Shape Optimization of Geometrically Nonlinear Structures

Using the level set method and topological derivatives, a topological shape optimization method that is independent of initial topology is developed for geometrically nonlinear structures in Total Lagrangian framework. In nonlinear topology optimization, response analysis may not converge due to relatively sparse material distribution driven by the conventional topology optimization such as homogenization and density methods. 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 both minimizing the objective function of instantaneous structural compliance and satisfying the required constraint of allowable material volume. In this paper, based on the obtained level set function, structural boundaries are actually represented in the response analysis. The developed method is able to create holes whenever and wherever necessary during the optimization and minimize the compliance through both shape and topological variations at the same time. The required velocity field in the initial domain to update the H-J equation is determined from the descent direction of Lagrangian derived from optimality conditions. The rest of velocity field is determined through a velocity extension method. Since the homogeneous material property and explicit boundary are utilized, the convergence difficulty is effectively prevented.