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Mesh analysis is a valuable method for simplifying circuit analysis using mesh currents as key circuit variables. Unlike nodal analysis, which focuses on determining unknown voltages, mesh analysis applies Kirchhoff's voltage law (KVL) to find unknown currents within a circuit. This method is particularly convenient in reducing the number of simultaneous equations that need to be solved.
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The fast decoupled power flow method addresses contingencies in power system operations, such as generator outages or transmission line failures. This method provides quick power flow solutions, essential for real-time system adjustments. Fast decoupled power flow algorithms simplify the Jacobian matrix by neglecting certain elements, leading to two sets of decoupled equations:
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Approximating inverse FEM matrices on non-uniform meshes with -matrices.

Niklas Angleitner1, Markus Faustmann1, Jens Markus Melenk1

  • 1Technische Universität Wien, Institute of Analysis and Scientific Computing (Inst. E 101), Wiedner Hauptstrasse 8-10, A-1040 Wien, Austria.

Calcolo
|November 22, 2021
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Summary
This summary is machine-generated.

This study demonstrates that the inverse of a finite element stiffness matrix can be efficiently approximated using hierarchical matrices ( H-matrices). This offers a data-sparse format for improved computational performance in solvers.

Keywords:
ApproximabilityFEMNon-uniform meshes

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Area of Science:

  • Numerical Analysis
  • Computational Mechanics
  • Scientific Computing

Background:

  • Finite element analysis (FEA) often involves large, dense stiffness matrices.
  • Direct solvers for FEA are computationally expensive due to matrix size and density.
  • Hierarchical matrices (H-matrices) offer a data-sparse representation for matrices.

Purpose of the Study:

  • To investigate the approximation of the inverse of the finite element stiffness matrix in the H-matrix format.
  • To establish the efficiency and accuracy of H-matrix approximation for stiffness matrices.
  • To explore the utility of H-matrix approximations as solvers or preconditioners.

Main Methods:

  • Consideration of shape-regular, potentially non-uniform meshes, including algebraically graded meshes.
  • Mathematical proof of approximation rates for the stiffness matrix inverse in H-matrix format.
  • Analysis of storage complexity and computational efficiency of H-matrix representations.

Main Results:

  • The inverse of the stiffness matrix can be approximated in the H-matrix format at an exponential rate concerning the block rank.
  • H-matrix storage complexity is logarithmic-linear and grows linearly with block rank.
  • Efficient approximations are achievable for a broad class of meshes.

Conclusions:

  • H-matrix approximation provides an efficient method for representing the inverse of finite element stiffness matrices.
  • These approximations are suitable for use as approximate direct solvers or preconditioners in iterative solvers.
  • The findings enable significant computational savings in FEA.