Implementation of continuum sensitivity analysis with existing finite element code

Park, Il-Han; Coulomb, J.L.; Hahn, Song–Yop

doi:10.1109/20.250752

Cited by 40 publications

(23 citation statements)

References 2 publications

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“…Assuming a discretization z = N p=1 z p w p of the unknown field, with basis functions w p ∈ Z 0 z and N the number of nodal unknowns, one solves with the direct method the linear problem (19), which is of the form…”

Section: Discussionmentioning

confidence: 99%

“…This term is therefore already known from the finite element solving, and needs not be recomputed when solving (19). The partial Lie derivative in (19) accounts for the explicit dependency (i.e. holding the field argument constant) of the residual on the variation of τ .…”

Section: Direct Approachmentioning

confidence: 99%

“…The Fréchet derivative term in (19) involves the tangent stiffness matrix of the nonlinear problem (1), and hence, in practice, the jacobian matrix of the problem after finite element discretization and convergence of the iterative nonlinear process. This term is therefore already known from the finite element solving, and needs not be recomputed when solving (19). The partial Lie derivative in (19) accounts for the explicit dependency (i.e.…”

Section: Direct Approachmentioning

confidence: 99%

“…For sensitivity calculations however, where the perturbation is infinitesimal, the EL approach is to be preferred since it offers the same accuracy at a much lower computational cost. In addition to bypassing the solution of the Laplace equation necessary for the LaS approach, the support of all the volume integrals in sensitivity equations (13), (19), (24) and (26) which involve the velocity field is reduced to the layer of elements connected to the moving interfaces of Number of elements Sensitivity FD-EL Lie-EL FD-LAS Lie-LAS Figure 5: A linear elastic model is defined over the plate, in Fig. 3, where the design variable τ is the major axis of the elliptical hole and the internal energy is the performance function.…”

Section: Design Velocity Field Computationmentioning

confidence: 99%

“…Sensitivity analysis has been developed so far mostly in the area of structural mechanics [10,11,12,1,13,14,15,16]. Applications in other disciplines such as electromagnetics have also been proposed [17,18,19], but have been limited to problems expressed in terms of a scalar potential, leaving aside the problem of handling vector unknown fields, which requires a more general theoretical framework. This generalized framework is the purpose of this paper.…”

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confidence: 99%

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Design sensitivity analysis for shape optimization based on the Lie derivative

Kuci

Henrotte

Duysinx

et al. 2017

Computer Methods in Applied Mechanics and Engineering

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The paper presents a theoretical framework for the shape sensitivity analysis of systems governed by partial differential equations. The proposed approach, based on geometrical concepts borrowed from differential geometry, shows that sensitivity of a performance function (i.e. any function of the solution of the problem) with respect to a given design variable can be represented mathematically as a Lie derivative, i.e. the derivative of that performance function along a flow representing the continuous shape modification of the geometrical model induced by the variation of the considered design variable. Theoretical formulae to express sensitivity analytically are demonstrated in detail in the paper, and applied to a nonlinear magnetostatic and a linear elastic problem, following both the direct and the adjoint approaches. Following the analytical approach, one linear system of which only the right-hand side needs be evaluated (the system matrix being known already) has to be solved for each of the design variables in the direct approach, or for each performance functions in the adjoint approach. A substantial gain in computation time is obtained this way compared to a finite difference evaluation of sensitivity, which requires solving a second nonlinear system for each design variable. This is the main motivation of the analytical approach. There is some freedom in the definition of the auxiliary flow that represents the shape modification. We present a method that makes benefit of this freedom to express sensitivity locally as a volume integral over a single layer of finite elements connected to both sides of the surfaces undergoing shape modification. All sensitivity calculations are checked with a finite difference in order to validate the analytic approach. Convergence is analyzed in 2D and 3D, with first and second order finite elements.

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Section: Discussionmentioning

confidence: 99%

Section: Direct Approachmentioning

confidence: 99%

Section: Direct Approachmentioning

confidence: 99%

Section: Design Velocity Field Computationmentioning

confidence: 99%

mentioning

confidence: 99%

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Design sensitivity analysis for shape optimization based on the Lie derivative

Kuci

Henrotte

Duysinx

et al. 2017

Computer Methods in Applied Mechanics and Engineering

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Introduction

Park¹

2018

Design Sensitivity Analysis and Optimization of Electromagnetic Systems

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Combination of topology optimization and Lie derivative-based shape optimization for electro-mechanical design

Kuci

Henrotte

Duysinx

et al. 2018

Struct Multidisc Optim

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This paper presents a framework for the simultaneous application of shape and topology optimization in electro-mechanical design problems. Whereas the design variables of a shape optimization are the geometrical parameters of the CAD description, the design variables upon which densitybased topology optimization acts represent the presence or absence of material at each point of the region where it is applied. These topology optimization design variables, which are called densities, are by essence substantial quantities. This means that they are attached to matter while, on the other hand, shape optimization implies ongoing changes of the model geometry. An appropriate combination of the two representations is therefore necessary to ensure a consistent design space as the joint shape-topology optimization process unfolds. The optimization problems dealt with in this paper are furthermore constrained to verify the governing partial differential equations (PDEs) of a physical model, possibly nonlinear and discretized by means of, e.g., the finite element method (FEM). Theoretical formulas, based on the Lie derivative, to express the sensitivity of the performance functions of the optimization problem are derived and validated to be used in gradient-based algorithms. The method is applied to the torque ripple minimization in an interior permanent magnet synchronous machine (PMSM).

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Implementation of continuum sensitivity analysis with existing finite element code

Cited by 40 publications

References 2 publications

Design sensitivity analysis for shape optimization based on the Lie derivative

Design sensitivity analysis for shape optimization based on the Lie derivative

Introduction

Combination of topology optimization and Lie derivative-based shape optimization for electro-mechanical design

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