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Constrained $L^p$ Approximation of Shape Tensors and its Role for the Determination of Shape Gradients

Laura Hetzel, Gerhard Starke

Abstract

This paper extends our earlier work [arXiv:2309.13595] on the $L^p$ approximation of the shape tensor by Laurain and Sturm. In particular, it is shown that the weighted $L^p$ distance to an affine space of admissible symmetric shape tensors satisfying a divergence constraint provides the shape gradient with respect to the $L^{p^\ast}$-norm (where $1/p + 1/p^\ast = 1$) of the elastic strain associated with the shape deformation. This approach allows the combination of two ingredients which have already been used successfully in numerical shape optimization: (i) departing from the Hilbert space framework towards the Lipschitz topology approximated by $W^{1,p^\ast}$ with $p^\ast > 2$ and (ii) using the symmetric rather than the full gradient to define the norm. Similarly to [arXiv:2309.13595], the $L^p$ distance measures the shape stationarity by means of the dual norm of the shape derivative with respect to the above-mentioned $L^{p^\ast}$-norm of the elastic strain. Moreover, the Lagrange multiplier for the momentum balance constraint constitute the steepest descent deformation with respect to this norm. The finite element realization of this approach is done using the weakly symmetric PEERS element and its three-dimensional counterpart, respectively. The resulting piecewise constant approximation for the Lagrange multiplier is reconstructed to a shape gradient in $W^{1,p^\ast}$ and used in an iterative procedure towards the optimal shape.

Constrained $L^p$ Approximation of Shape Tensors and its Role for the Determination of Shape Gradients

Abstract

This paper extends our earlier work [arXiv:2309.13595] on the approximation of the shape tensor by Laurain and Sturm. In particular, it is shown that the weighted distance to an affine space of admissible symmetric shape tensors satisfying a divergence constraint provides the shape gradient with respect to the -norm (where ) of the elastic strain associated with the shape deformation. This approach allows the combination of two ingredients which have already been used successfully in numerical shape optimization: (i) departing from the Hilbert space framework towards the Lipschitz topology approximated by with and (ii) using the symmetric rather than the full gradient to define the norm. Similarly to [arXiv:2309.13595], the distance measures the shape stationarity by means of the dual norm of the shape derivative with respect to the above-mentioned -norm of the elastic strain. Moreover, the Lagrange multiplier for the momentum balance constraint constitute the steepest descent deformation with respect to this norm. The finite element realization of this approach is done using the weakly symmetric PEERS element and its three-dimensional counterpart, respectively. The resulting piecewise constant approximation for the Lagrange multiplier is reconstructed to a shape gradient in and used in an iterative procedure towards the optimal shape.
Paper Structure (6 sections, 6 theorems, 96 equations, 4 figures, 2 tables)

This paper contains 6 sections, 6 theorems, 96 equations, 4 figures, 2 tables.

Table of Contents

  1. Shape tensor formulation and shape gradients with respect to elastic strain norms
  2. Constrained $L^p$ best approximation of shape tensors and its relation to $W^{1,p^\ast}$ shape gradients
  3. Discretization with weakly symmetric tensor finite elements
  4. Shape gradient iteration with reconstructed deformations in $W^{1,p^\ast} (\Omega)$
  5. Adjustment of the rigid body modes
  6. Extension to Problems with Constraints on the Shape

Key Result

Theorem 2.1

Let $p \in ( 1 , 2 ]$ and assume that $K ( u_\Omega , y_\Omega ) \in L^p ( \Omega ; \mathds{S}^{d \times d} )$. If the compatibility condition $( f \: \nabla y_\Omega , \rho ) = 0$ for all $\rho \in {\rm RM} (\mathds{R}^d)$ is fulfilled, then there is a uniquely determined $L^p (\Omega)$-best approx

Figures (4)

  • Figure 1: Example 1: Polygonal domains approximating a disk
  • Figure 2: Example 1: Values $\eta_{1.1,h} (\Omega_h)$ for square, octagon and hexadecagon
  • Figure 3: Result of shape gradient iteration for $p = 2$ (left) and $p = 1.1$ (right)
  • Figure 4: Final shapes for $p = 2$ (left) and $p = 1.1$ (right) on levels 4 (top) and 5 (bottom)

Theorems & Definitions (15)

  • Theorem 2.1
  • proof
  • Remark 2.2
  • Theorem 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • proof
  • ...and 5 more