Exact integration scheme for planewave-enriched partition of unity finite element method to solve the Helmholtz problem

In this paper, we present an exact integration scheme to compute highly oscillatory integrals that appear in the solution of the two-dimensional Helmholtz problem using the planewave-enriched partition of unity finite element method. In the proposed scheme, such oscillatory integrals are computed by...

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Bibliographic Details
Published in:Computer methods in applied mechanics and engineering 2017-04, Vol.317, p.619-648
Main Authors: Banerjee, Subhajit, Sukumar, N.
Format: Article
Language:English
Subjects:
Divergence
Divergence theorem
Exact integration
Finite element analysis
Finite element method
Helmholtz equation
Helmholtz equations
Highly oscillatory integrals
Integrals
Mathematical analysis
Partition of unity finite element method
Partitions
Planewave enrichment
Polynomials
Studies
Unity
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Summary:In this paper, we present an exact integration scheme to compute highly oscillatory integrals that appear in the solution of the two-dimensional Helmholtz problem using the planewave-enriched partition of unity finite element method. In the proposed scheme, such oscillatory integrals are computed by a recursive application of the divergence theorem, eventually expressing the integrals in terms of evaluations of the corresponding integrands at the nodes of the finite element mesh. The number of such function evaluations is independent of the wave number k, which permits the scheme to be used for arbitrary high values of k. We consider finite element meshes with unstructured triangular and structured rectangular elements, and present numerical results for three canonical benchmark Helmholtz problems to demonstrate the accuracy and efficacy of the method. •Recursive use of Gauss’s theorem for integration of polynomial-exponential products on polygons.•Integration scheme provides exact results for any wave number k.•Exact computation of stiffness matrix entries in the planewave-enriched PUFEM.•Method is efficient on unstructured and structured meshes for Helmholtz problems.
ISSN:0045-7825
1879-2138
DOI:10.1016/j.cma.2017.01.001