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Boundary Element Method Introduction
Stephen Kirkup School of Engineering, University of Central Lancashire, England
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Purpose The Boundary Element Method is a Numerical Method
From a mathematical viewpoint the BEM finds an approximate solution to a partial differential equation (PDE) governing a domain For an engineer the BEM can be used to simulate the properties of a design before committing the building the prototyp Domain of ‘just about any reasonable’ shape and general conditions Analytic methods only work for simple separable geometries - eg circles, squares, spheres and cylinders. Useful for generating test problems.
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Boundary Value Problem - Boundary
The BEM solves a PDE a boundary value problem Domain is OR
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Boundary Value Problem - Conditions
The BEM solves a PDE with boundary conditions; a boundary value problem 𝜕𝜑 𝜕𝑛 =𝑓 α 𝜑+β 𝜕𝜑 𝜕𝑛 =𝑓 𝜑=𝑓 Dirichlet or Essential Neuman or flux Robin eg impedance Note: Exterior problems also have a condition at ∞, that is in the far-field Game is : Given the PDE, the boundary and the boundary condition, we need to find 𝜑 in the domain To solve the boundary value problem
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Motivation Finite element method and the finite difference method?
FDM: 27 internal nodes FEM: 42 elements BEM: 16 elements Finer Mesh? BEM is possibly more efficient to scale 3D? Only the surface needs to be meshed – benefits of BEM seem more pronounced Ease of use? Boundary mesh is ‘easier’ or ‘less bother’ to set up than a domain mesh
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Motivation Motivation - FEM for exterior problems
Possible Solution – Infinite elements
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Motivation Motivation - FDM for exterior problems
Possible Solution – Perfectly matched layer
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Motivation - BEM for exterior problems
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Initial Summary In the boundary element method only a boundary mesh is required Hence possibly a lot less elements or nodes – possible efficiency saving Possible reduction in run time Less ‘bother’ in comparison to the other methods Saves development time for the eventual user – typically an engineer Raises the profile of the BEM as a teaching package An important research area in humble SK’s view and one that has been largely overlooked Requires a reformulation of the PDE as a boundary-only equation Can be done for some linear elliptic PDEs Quite a restriction although some progress with DRM/MRM
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BEM in Practice Is used in most areas of physics/engineering, but not as widely as FEM Can be thought of as a competitor to the FEM or FDM More realistic to see it as a complement to these methods Could be thought of as an alternative to PML or infinite elements for exterior problems In a ‘multiphysics’ situation, the preferred method could be used in each domain eg fluid-structure BEM-FEM BEM FEM/FDM BEM FEM
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Wot? Still not much maths?
Where are we coming from? Mathematician or Engineer? Researcher, Teacher or Student? How do I learn about the method? Mathematical basis of BEM is complicated Traditionally, books on BEM are impenetrable Educational viewpoint in SK’s humble opinion Work towards making BEM theory accessible and inclusive Easy to use the method and so educational possiblities within and outside of BEM world
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This week Develop the BEM for the solution of Laplace’s equation
Simplest equation to which the BEM can be applied Is widely applicable in physics and engineering: ‘potential’ problems The model equation for the BEM Most problems you find with the application of the BEM to any other problem area will be found in this one It is the best equation with which we learn and teach the BEM and …. Develop the BEM for the solution of the Helmholtz Equation Acoustics Eigenvalue Problems Fluid-Structure Interaction
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