Analytic element method
Encyclopedia
The analytic element method (AEM) is a numerical
method used for the solution of partial differential equation
s. It was initially developed by O.D.L. Strack at the University of Minnesota
. It is similar in nature to the boundary element method
(BEM), as it does not rely upon discretization of volumes or areas in the modeled system; only internal and external boundaries are discretized. One of the primary distinctions between AEM and BEMs is that the boundary integrals are calculated analytically.
The analytic element method is most often applied to problems of groundwater flow
governed by the Poisson equation, though it is applicable to a variety of linear partial differential equations, including the Laplace, Helmholtz
, and biharmonic
equations.
The basic premise of the analytic element method is that, for linear differential equation
s, elementary solutions may be superimposed to obtain more complex solutions. A suite of 2D and 3D analytic solutions ("elements") are available for different governing equations. These elements typically correspond to a discontinuity in the dependent variable or its gradient along a geometric boundary (e.g., point, line, ellipse, circle, sphere, etc.). This discontinuity has a specific functional form (usually a polynomial in 2D) and may be manipulated to satisfy Dirichlet, Neumann, or Robin (mixed) boundary conditions. Each analytic solution is infinite in space and/or time. In addition, each analytic solution contains degrees of freedom (coefficients) that may be calculated to meet prescribed boundary conditions along the element's border. To obtain a global solution (i.e., the correct element coefficients), a system of equations is solved such that the boundary conditions are satisfied along all of the elements (using collocation
, least-squares minimization
, or a similar approach). Notably, the global solution provides a spatially continuous description of the dependent variable everywhere in the infinite domain, and the governing equation is satisfied everywhere exactly except along the border of the element, where the governing equation is not strictly applicable due to the discontinuity.
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
method used for the solution of partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
s. It was initially developed by O.D.L. Strack at the University of Minnesota
University of Minnesota
The University of Minnesota, Twin Cities is a public research university located in Minneapolis and St. Paul, Minnesota, United States. It is the oldest and largest part of the University of Minnesota system and has the fourth-largest main campus student body in the United States, with 52,557...
. It is similar in nature to the boundary element method
Boundary element method
The boundary element method is a numerical computational method of solving linear partial differential equations which have been formulated as integral equations . It can be applied in many areas of engineering and science including fluid mechanics, acoustics, electromagnetics, and fracture...
(BEM), as it does not rely upon discretization of volumes or areas in the modeled system; only internal and external boundaries are discretized. One of the primary distinctions between AEM and BEMs is that the boundary integrals are calculated analytically.
The analytic element method is most often applied to problems of groundwater flow
Groundwater flow equation
Used in hydrogeology, the groundwater flow equation is the mathematical relationship which is used to describe the flow of groundwater through an aquifer. The transient flow of groundwater is described by a form of the diffusion equation, similar to that used in heat transfer to describe the flow...
governed by the Poisson equation, though it is applicable to a variety of linear partial differential equations, including the Laplace, Helmholtz
Helmholtz equation
The Helmholtz equation, named for Hermann von Helmholtz, is the elliptic partial differential equation\nabla^2 A + k^2 A = 0where ∇2 is the Laplacian, k is the wavenumber, and A is the amplitude.-Motivation and uses:...
, and biharmonic
Biharmonic equation
In mathematics, the biharmonic equation is a fourth-order partial differential equation which arises in areas of continuum mechanics, including linear elasticity theory and the solution of Stokes flows...
equations.
The basic premise of the analytic element method is that, for linear differential equation
Linear differential equation
Linear differential equations are of the formwhere the differential operator L is a linear operator, y is the unknown function , and the right hand side ƒ is a given function of the same nature as y...
s, elementary solutions may be superimposed to obtain more complex solutions. A suite of 2D and 3D analytic solutions ("elements") are available for different governing equations. These elements typically correspond to a discontinuity in the dependent variable or its gradient along a geometric boundary (e.g., point, line, ellipse, circle, sphere, etc.). This discontinuity has a specific functional form (usually a polynomial in 2D) and may be manipulated to satisfy Dirichlet, Neumann, or Robin (mixed) boundary conditions. Each analytic solution is infinite in space and/or time. In addition, each analytic solution contains degrees of freedom (coefficients) that may be calculated to meet prescribed boundary conditions along the element's border. To obtain a global solution (i.e., the correct element coefficients), a system of equations is solved such that the boundary conditions are satisfied along all of the elements (using collocation
Collocation
In corpus linguistics, collocation defines a sequence of words or terms that co-occur more often than would be expected by chance. In phraseology, collocation is a sub-type of phraseme. An example of a phraseological collocation is the expression strong tea...
, least-squares minimization
Least squares
The method of least squares is a standard approach to the approximate solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in solving every...
, or a similar approach). Notably, the global solution provides a spatially continuous description of the dependent variable everywhere in the infinite domain, and the governing equation is satisfied everywhere exactly except along the border of the element, where the governing equation is not strictly applicable due to the discontinuity.