Shape Design Sensitivity Analysis Research Articles

Accurate and efficient boundary method for isogeometric shape design sensitivity analysis considering tangential divergence of non-smooth boundary

Accurate and efficient boundary method for isogeometric shape design sensitivity analysis considering tangential divergence of non-smooth boundary

Sensitivity analysis and design optimization of 3T rotating thermoelastic structures using IGBEM

<abstract> <p>In this study, the isogeometric boundary element method (IGBEM) based on non-uniform rational basis spline (NURBS) is used to perform shape design sensitivity and optimization of rotating three-temperature (3T) thermoelastic structures. During the optimization process, the shape design sensitivity within the IGBEM formulation was derived to include precise geometries and greater continuities. It was found through the application of the IGBEM that the shape design velocity has a significant effect on accuracy of the obtained shape design sensitivity. As a result, the developed shape design sensitivity analysis (SDSA) technique based on the considered IGBEM formulation outperforms the computational solution based on the traditional SDSA method. The isogeometric shape sensitivity and optimal design for a complicated three-temperature thermoelastic problem in rotating structures are investigated. The impact of rotation on the thermal stress sensitivity, optimal three-temperature, optimal displacement and optimal three temperature thermal stress distributions are established. It is shown that the SDSA derived using IGBEM is efficient and applicable for most three-temperature thermoelastic optimization problems.</p> </abstract>

Shape design optimization of interdigitated electrodes for maximal electro-adhesion forces

We present a shape design optimization method for interdigitated electrodes in an electro-adhesive device. In the finite element analysis of electrostatics using linear basis functions, a finite node displacement method is used for the accurate electro-adhesive force by integrating the electric field along the boundary surface. The floating potential boundary for conductive objects is handled by a charge conservation law in the electrostatic analysis as well as the shape design sensitivity analysis. In numerical examples, the structural shape of interdigitated electrodes is optimized to maximize the electro-adhesion force per unit area for both conductive and non-conductive objects. It turns out that the electro-adhesive force is mainly induced by an electrostatic induction for conductive objects and by an electric polarization for non-conductive ones. There is an optimal ratio of electrode width and air gap thickness for non-conductive objects but no limit for conductive objects.

Shape sensitivity analysis for non-differentiable performance measures in multiscale crack propagation simulation using regression hybrid method

ABSTRACTIn this paper, we present a regression hybrid method that calculates shape sensitivity coefficients for multiscale crack propagation problems with performance measures that are non-differentiable in numerical implementation. These measures are crack propagation speed (or crack speed) defined at atomistic level obtained by solving coupled atomistic/continuum structures using the bridging scale method (BSM). The major contributions of this paper are: first, by analyzing the characteristics of the performance measures of crack speed in design space, this paper verifies for the first time that these measures are theoretically continuous and differentiable with respect to design variables, and as a result, the sensitivity coefficients exist in theory; second, to overcome the non-differentiability of the performance measures in numerical computation due to the finite size of integration time step, this paper proposes a regression hybrid method that calculates the shape sensitivity coefficients of crack speed through polynomial regression analysis based on the sensitivity of atomic responses, which is calculated through analytical shape design sensitivity analysis (DSA). And finally, the proposed method supports for 3D crack propagation problems with periodic boundary condition in one direction. A nano-beam example is used to demonstrate numerically the feasibility, accuracy, and efficiency of the proposed method.

Isogeometric shape design sensitivity analysis of elasticity problems using boundary integral equations

Using boundary integral equations and isogeometric approach, a shape design sensitivity analysis (DSA) method is developed for two dimensional elastic structures. In the isogeometric approach, NURBS basis functions in CAD systems are directly utilized in response analysis, which enables a seamless incorporation of exact geometry and higher continuity into computational framework. To enhance the accuracy of shape design sensitivity, the CAD-based higher-order geometric information such as curvature, normal, and tangential vector is exactly embedded in the sensitivity expressions. In boundary integral formulation, shape design velocity field is decomposed into normal and tangential components, which significantly affect the accuracy of shape design sensitivity. Also, the proposed boundary-based method does not require the tedious design parameterization of internal domain. Through the numerical examples, the developed shape DSA method turns out to be more accurate than conventional finite element based one.

Adjoint shape design sensitivity analysis of molecular dynamics for lattice structures using GLE impedance forces

We presented a shape design sensitivity analysis method for lattice structures using a generalized Langevin equation (GLE) to overcome the difficulty of discrete nature in atomic systems. Taking advantage of the GLE forces, the perturbed atomistic region is treated as the GLE impedance forces and the shape design problem of discrete atomic variations is converted into a non-shape problem with GLE impedance forces. We developed an adjoint variable method in order to improve the computational efficiency for molecular dynamics (MD) with many design variables. Due to the translational symmetry in lattice structure, the size of the time history kernel function that accounts for the boundary effects of reduced systems could be reduced to that of a single atoms DOFs. In numerical examples, the convergent characteristic of shape sensitivity according to the amount of shape variations is investigated in MD systems. Also, the results of the derived shape sensitivity turn out to be more accurate and efficient, compared with those of the finite difference ones.

Shape Design Optimization of Crack Propagation ProblemsUsing 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.본 논문에서는 재생 커널 기법을 사용하여 혼합모드 균열진전 문제에 대한 연속체 기반의 형상 설계민감도 해석을 수행하였다. 재생 커널 기법은 기존의 유한요소법과 달리 요소망을 재구성할 필요가 없어, 커널 함수의 연속성을 증가시켰을 때 높은 정밀도의 형상함수를 얻을 수 있다는 장점을 가지고 있다. 균열선단 주변에서 J-적분을 수행하기 위해 선형탄성 조건이 고려되었다. 변위장과 응력 확대 계수의 설계변수에 대한 감도해석을 위하여 물질도함수를 도입하였으며 직접 미분법보다 효율적인 애조인 방법을 사용하여 설계민감도를 유도하였다. 수치 예제들을 통해서 재생 커널 기법을 이용한 균열진전 해석 결과의 타당성을 확인하였으며 애조인 방법을 이용한 형상 설계민감도 해석 결과를 유한차분법과 비교하여 매우 정확하고 효율적인 결과를 얻을 수 있음을 알 수 있었다. 이를 바탕으로 간단한 모델에 대하여 형상 최적설계를 수행하여 균열이 발생될 수 있는 구조물에 대해서 균열에 의한 피해를 최소화할 수 있도록 균열을 제어할 수 있는 최적의 형상을 도출하였다.

Shape Design Sensitivity Analysis of Dynamic Crack Propagation Problems using Peridynamics and Parallel Computation

Using the bond-based peridynamics and the parallel computation with binary decomposition, an adjoint shape design sensitivity analysis(DSA) method is developed for the dynamic crack propagation problems. The peridynamics includes the successive branching of cracks and employs the explicit scheme of time integration. The adjoint variable method is generally not suitable for path-dependent problems but employed since the path of response analysis is readily available. The accuracy of analytical design sensitivity is verified by comparing it with the finite difference one. The finite difference method is susceptible to the amount of design perturbations and could result in inaccurate design sensitivity for highly nonlinear peridynamics problems with respect to the design. It turns out that C1-continuous volume fraction is necessary for the accurate evaluation of shape design sensitivity in peridynamic discretization.

Isogeometric shape design sensitivity analysis of stress intensity factors for curved crack problems

An isogeometric shape design sensitivity analysis method is developed for the stress intensity factors (SIFs) in curved crack problems. In order to obtain an enhanced shape sensitivity of SIFs, exact higher-order geometric information of curvature and normal vector is embedded in the isogeometric shape sensitivity expressions. A direct differentiation method is employed for the design sensitivity analysis and the size and orientation of the crack are selected as design variables. In the isogeometric approach, the NURBS basis functions in CAD system are directly utilized in the response analysis, which enables a seamless incorporation of exact geometry and higher continuity into the computational framework. The precise evaluation of the crack-face integral as well as the interaction integral is essential for the computation of the stress intensity factors for the curved crack problems in a mixed-mode. The CAD-based exact representation of tangential and normal vectors enables us to exactly define a local coordinate system at the crack-tip, whose shape dependency naturally leads to configuration design variations that include the change of crack orientation. Compared with the conventional finite element approach, a higher continuity of stress and strain fields are expected in the interaction integral domain. The design dependencies of auxiliary field and q-function are considered in the design sensitivity formulation, reflecting the convection effects due to the non-constant design velocity in the interaction integral domain. Various numerical examples of curved crack problems are presented to verify the developed isogeometric sensitivity analysis method through the comparison with exact and finite element solutions.

Adjoint shape design sensitivity analysis of fluid–solid interactions using concurrent mesh velocity in ALE formulation

A coupled variational equation for fluid–solid interaction (FSI) problems is derived using a steady state Navier–Stokes equation for incompressible flows, an equilibrium equation for geometrically nonlinear solids, a traction continuity condition at interfaces, and a pseudo-equilibrium equation for mesh velocity. The moving boundary in arbitrary Lagrangian–Eulerian (ALE) formulation is included in the variational equations by the mesh velocity obtained from a displacement-loaded pseudo-structural problem at a concurrent configuration, which eventually facilitates to derive shape design sensitivity. A continuum-based adjoint shape sensitivity is derived under ALE formulation, which turns out to be very accurate and efficient due to the utilization of converged tangent and the linearity of both adjoint and sensitivity equations. Through numerical examples, the obtained sensitivity is verified in terms of accuracy and efficiency compared with finite difference sensitivity and further applied to the shape optimization problem of finding a stiff structure while satisfying a volume constraint.

사판식 구동모터에 장착된 밸브의 설계변수 민감도 해석 사례

본 논문은 컴퓨터 해석프로그램인 SimulationX를 이용하여 굴삭기 주행모터의 내부에서 발생하는 서지압력의 저감 방법에 대하여 분석하는 연구이다. 설계민감도 해석을 통하여 설계상의 문제점을 파악하고 해결책에 접근하는 방법을 다룬다. 진행순서는 다음과 같다. 우선 현재 설계 된 주행모터에 장착된 밸브들의 동적거동을 분석하여 설계의 문제점을 찾아낸다. 그 후 많은 설계변수들 중 동적성능향상에 민감하다고 판단되는 변수를 도출하고 설계치수를 조정하여 동적성능 안정화의 경향을 살펴본다. 마지막으로 민감한 변수가 다수 일 경우 변수조합을 통해 효과적인 튜닝방법을 제안한다. This paper is about study how to decrese surge pressure that is occurred in excavator driving motor. We used computer simulation program SimulationX. It is also about the way finding design problem and approaching a solution through interpreting shape design sensitivity analysis. Programmes are below. First of all, finding shape fault by analyzing dynamic behavior of valves installed in hydraulic driving motor which is designed now. And drawing variable which is considered sensitive to improve dynamic efficiency among a lot of shape variables. Then, targeting that variable and examining dynamic efficiency stabilization tendency with controlling it. Finally, suggesting the most effective tuning method through variable combination as there are a lot of sensitive variables.

민들린 평판의 아이소-지오메트릭 형상 설계민감도 해석

Abstract In this paper, a shape design sensitivity analysis(DSA) method is presented for Mindlin plates using an isogeometric approach. The isogeometric method possesses desirable advantages; the representation of exact geometry and the higher order inter-element continuity, which lead to the fast convergence of solution as well as accurate sensitivity results. Unlike the finite element methods using linear shape functions, the isogeometric method considers the exact normal vector and curvature of the CAD geometry, taking advantages of higher order NURBS basis functions. A selective reduced integration(SRI) technique is incorporated to overcome the difficulty of 'shear locking' phenomenon. This simple technique is surprisingly helpful for the accuracy of the isogeometric shape sensitivity without complicated formulation. Through the numerical examples of plate bending problems, the accuracy of the proposed isogeometric analysis method is compared with that of finite element one. Also, the isogeometric shape sensitivity turns out to be very accurate when compared with finite difference sensitivity.

Equivalence of continuum and discrete analytic sensitivity methods for nonlinear differential equations

This paper presents results for the study of equivalence between the total form continuum sensitivity equation (CSE) method and the discrete analytic method of shape design sensitivity analysis. For the discrete analytic method, the sensitivity equations are obtained by taking analytic derivatives of the discretized equilibrium equations with respect to the shape design parameters. For the CSE method, the equilibrium equations are firstly differentiated to form a set of linear continuous sensitivity equations and then discretized for solving the shape sensitivities. The sensitivity equations can be derived by taking the material derivatives (total form) or the partial derivatives (local form) of the equilibrium equations. The total form CSE method is shown for the first time to be equivalent, after finite element discretization, to the discrete analytic method for nonlinear second-order differential equations of a particular form with design dependent loads when they use the same: (1) finite element discretization, (2) numerical integration of element matrices, (3) design velocity fields that are linear with respect to the design variable and (4) shape functions for domain transformation and for design velocity field calculations. The shape sensitivity equations for three-dimensional geometric nonlinear elastic structures and linear potential flow are derived by using both total form CSE and discrete analytic method to show the equivalence of the two methods for these specific examples. The accuracy of shape sensitivity analysis is verified by potential flow around an airfoil and a joined beam with an airfoil under gust load. The results show that analytic sensitivity results are consistent with the complex step results.

일반 곡면 좌표계에서 구현된 아이소-지오메트릭 형상 설계민감도 해석

Finite element analysis is to approximate a geometry model developed in computer-aided design(CAD) to a finite element model, thus the conventional shape design sensitivity analysis and optimization using the finite element method have some difficulties in the parameterization of geometry. However, isogeometric analysis is to build a geometry model and directly use the functions describing the geometry in analysis. Therefore, the geometric properties can be embedded in the NURBS basis functions and control points so that it has potential capability to overcome the aforementioned difficulties. In this study, the isogeometric structural analysis and shape design sensitivity analysis in the generalized curvilinear coordinate(GCC) systems are discussed for the curved geometry. Representing the higher order geometric information, such as normal, tangent and curvature, yields the isogeometric approach to be the best way for generating exact GCC systems from a given CAD geometry. The developed GCC isogeometric structural analysis and shape design sensitivity analysis are verified to show better accuracy and faster convergency by comparing with the results obtained from the conventional isogeometric method.

Isogeometric shape design sensitivity analysis using transformed basis functions for Kronecker delta property

The isogeometric shape design sensitivity analysis (DSA) includes the desirable features; easy design parameterization and accurate shape sensitivity embedding the higher-order geometric information of curvature and normal vector. Due to the non-interpolatory property of NURBS basis, however, the imposition of essential boundary condition is not so straightforward in the isogeometric method. Taking advantages of geometrically exact property, an isogeometric DSA method is developed applying a mixed transformation to handle the boundary condition. A set of control point and NURBS basis function is added using the h-refinement and Newton iterations to precisely locate the control point to impose the boundary condition. In spite of additional transformation, its computation cost is comparable to the original one with penalty approach since the obtained Kronecker delta property enables to reduce the size of system matrix. Through demonstrative numerical examples, the effectiveness, accuracy, and computing cost of the developed DSA method are discussed.

Design of rotor shape to reduce torque ripple in IPM motors

This paper presents a shape design sensitivity analysis of the effect of geometrical parameters on torque pulsations of an interior permanent magnet (IPM) synchronous motor. The analysis was conducted in two dimensions using the finite element method (FEM). Calculation results have been compared with experimental results.

Design sensitivity analysis and optimization of interface shape for zoned-inhomogeneous thermal conduction problems using boundary integral formulation

A generalized formulation of the shape design sensitivity analysis for two-dimensional steady-state thermal conduction problem as applied to zoned-inhomogeneous solids is presented using the boundary integral and the adjoint variable method. Shape variation of the external and zone-interface boundary is considered. Through an analytical example, it is proved that the derived sensitivity formula coincides with the analytic solution. In numerical implementation, the primal and adjoint problems are solved by the boundary element method. Shape sensitivity is numerically analyzed for a compound cylinder, a thermal diffuser and a cooling fin problem, and its accuracy is compared with that by numerical differentiation. The sensitivity formula derived is incorporated to a nonlinear programming algorithm and optimum shapes are found for the thermal diffuser and the cooling fin problem.

Numerical method for shape optimization using T-spline based isogeometric method

Numerical methods for shape design sensitivity analysis and optimization have been developed for several decades. However, the finite-element-based shape design sensitivity analysis and optimization have experienced some bottleneck problems such as design parameterization and design remodeling during optimization. In this paper, as a remedy for these problems, an isogeometric-based shape design sensitivity analysis and optimization methods are developed incorporating with T-spline basis. In the shape design sensitivity analysis and optimization procedure using a standard finite element approach, the design boundary should be parameterized for the smooth variation of the boundary using a separate geometric modeler, such as a CAD system. Otherwise, the optimal design usually tends to fall into an undesirable irregular shape. In an isogeometric approach, the NURBS basis function that is used in representing the geometric model in the CAD system is directly used in the response analysis, and the design boundary is expressed by the same NURBS function as used in the analysis. Moreover, the smoothness of the NURBS can allow the large perturbation of the design boundary without a severe mesh distortion. Thus, the isogeometric shape design sensitivity analysis is free from remeshing during the optimization process. In addition, the use of T-spline basis instead of NURBS can reduce the number of degrees of freedom, so that the optimal solution can be obtained more efficiently while yielding the same optimum design shape.

Shape Optimization Based on the Local Natural Neighbor Petrov-Galerkin Method

Based on the material derivative concept and direct differentiation approach,a numerical method for shape design sensitivity analysis using meshless natural neighbor Petrov-Galerkin method (NNPG) is proposed. The local weak form of governing equation of NNPG is directly differentiated with respect to design variables and subsequently discretized with NNPG to obtain the sensitivities. Based on the mathematical programming method,a shape optimization method is proposed,and the natural neighbour Petrov-Galerkin method is used to calculate the structure responses and their sensitivities. The numerical example shows the proposed optimization method is totally automatic,efficient. Absolutely,no remeshing is needed.

Sensitivity analysis and shape optimization based on FE–EFG coupled method

By use of 4-node isoparametric quadrangle interface element between finite element (FE) and meshless regions, a collocation approach is introduced to couple firstly FE and element-free Galerkin (EFG) method in this paper. By taking derivative of discreteness equilibrium equation at interface element with respect to design variable, a numerical method for discreteness-based shape design sensitivity analysis in interface element is obtained. The design sensitivity analysis (DSA) of coupled FE–EFG method is achieved by employing the DSA of nodal displacement at the interface element. The numerical method presented is testified by examples. It can be observed excellent agreement between the numerical results and the analytical solution. Finally the shape optimization of fillet is achieved by using coupled FE–EFG method. The result obtained show that imposing of the essential boundary condition is easy to implement, the computational time is reduced and the distortion of mesh is avoided.