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Interface Identification constrained by Local-to-Nonlocal Coupling

Matthias Schuster, Volker Schulz

TL;DR

This work makes use of an energy-based Local-to-Nonlocal coupling that serves as a constraint for an interface identification problem and combines partial differential operators with nonlocal operators.

Abstract

Models of physical phenomena that use nonlocal operators are better suited for some applications than their classical counterparts that employ partial differential operators. However, the numerical solution of these nonlocal problems can be quite expensive. Therefore, Local-to-Nonlocal couplings have emerged that combine partial differential operators with nonlocal operators. In this work, we make use of an energy-based Local-to-Nonlocal coupling that serves as a constraint for an interface identification problem.

Interface Identification constrained by Local-to-Nonlocal Coupling

TL;DR

This work makes use of an energy-based Local-to-Nonlocal coupling that serves as a constraint for an interface identification problem and combines partial differential operators with nonlocal operators.

Abstract

Models of physical phenomena that use nonlocal operators are better suited for some applications than their classical counterparts that employ partial differential operators. However, the numerical solution of these nonlocal problems can be quite expensive. Therefore, Local-to-Nonlocal couplings have emerged that combine partial differential operators with nonlocal operators. In this work, we make use of an energy-based Local-to-Nonlocal coupling that serves as a constraint for an interface identification problem.
Paper Structure (14 sections, 13 theorems, 93 equations, 5 figures, 2 algorithms)

This paper contains 14 sections, 13 theorems, 93 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Nonlocal Terminology and Framework
  3. Interface Identification constrained by an energy-based Local-to-Nonlocal Coupling
  4. Energy-based Local-to-Nonlocal Coupling
  5. Schwarz Methods
  6. Problem Formulation
  7. Shape Optimization
  8. Basics of Shape Optimization
  9. Shape Derivative via the Averaged Adjoint Method
  10. Deriving the Shape Derivative of the Reduced Functional
  11. Optimization Algorithm
  12. Numerical Experiments
  13. Conclusion
  14. Proof of Lemma \ref{['lemma:AAM_reduced_functional']}

Key Result

Lemma 3.5

Let $\gamma$ fulfill (K1)-(K4). Then, the space $(H,<\cdot,\cdot\cdot>_{H})$ is a Hilbert space and the norm $||\cdot||_{H}$ is equivalent to the norm of the product space $\left(H^1(\Omega_l), ||\cdot||_{H^1(\Omega_l)}\right)\times \left(L_c^2(\Omega_{nl} \cup \mathcal{I}_{nl}), ||\cdot||_{L^2(\Ome

Figures (5)

  • Figure 3.1: Here, the nonlocal domain $\Omega_{nl}$ is depicted in the darker red and the corresponding nonlocal domain $\mathcal{I}_{nl}$ of the nonlocal domain $\Omega_{nl}$ is colored in the lighter red. Moreover, the 'local' domain $\Omega_l$ consists in this example of the nonlocal boundary $\mathcal{I}_{nl}$ as well as the gray area.
  • Figure 4.1: The perturbation of identity moves a point $\mathbf{x}$ in the direction of a vector field $\mathbf{V}$ to the point $\mathbf{F}_{\mathbf{t}}(\mathbf{x})$. Applied on $\Gamma$, this interface is shifted and deformed to a new interface $\mathbf{F}_{\mathbf{t}}(\Gamma)$.
  • Figure 5.1: Example 1
  • Figure 5.2: Example 2
  • Figure 5.3: On the left hand side we see the development of the objective functional values during the algorithm for each example. Here, the starting solution of the first example is quite near the optimal solution such that the objective function value decreases only a bit. In the second example the objective function value is reduced quite fast in the first twelve iterations. After that there is also only a small improvement regarding the objective function value. On the right hand side, we can see the history of the $L^2(\Omega)$ norm of the shape gradients.

Theorems & Definitions (34)

  • Definition 2.1
  • Example 2.2
  • Remark 3.1
  • Definition 3.2
  • Definition 3.3
  • Remark 3.4
  • Lemma 3.5: After AcostaDD
  • proof
  • Corollary 3.6
  • Theorem 3.7: AcostaDD
  • ...and 24 more