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Force Computation for Dielectrics using Shape Calculus

by P. Panchal and N. Ren and R. Hiptmair

(Report number 2022-21)

Abstract
We are concerned with the numerical computation of electrostatic forces/torques in only piecewise homogeneous materials using the boundary element method (BEM). Conventional force formulas based on the Maxwell stress tensor yield functionals that fail to be continuous on natural trace spaces. Thus their use in conjunction with BEM incurs slow convergence and low accuracy. We employ the remedy discovered in \([\){\sc P.~Panchal and R.~Hiptmair}, {\em Electrostatic {Force} {Computation} with {Boundary} {Element} {Methods}}, The SMAI journal of computational mathematics, 8 (2022), pp.~49--74\(]\): Motivated by the virtual work principle which is interpreted using techniques of shape calculus, and using the the adjoint method from shape optimization, we derive stable interface-based force functionals suitable for use with BEM. This is done in the framework of single-trace direct boundary integral equations for second-order transmission problems. Numerical tests confirm the fast asymptotic convergence and superior accuracy of the new formulas for the computation of total forces and torques.

Keywords: Electrostatics, Electrostatic forces, Shape derivative, Boundary integral equations, Boundary element method, Virtual Work Principle, Dielectric, Single trace formulation

@Techreport{PRH22_1009,
  author = {P. Panchal and N. Ren and R. Hiptmair},
  title = {Force Computation for Dielectrics using Shape Calculus},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2022-21},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2022/2022-21.pdf },
  year = {2022}
}

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