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# Analytic element method

The analytic element method (AEM) is a numerical method used for the solution of partial differential equations.[1][2][3] 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.

Flow around impermeable cylinders. Solved with the AEM using 20 coefficients in the series expansions.

## Mathematical basis

The basic premise of the analytic element method is that, for linear differential equations, 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.

Commonly 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.

The ability to superpose numerous elements in a single solution means that analytical solutions can be realized for arbitrarily complex boundary conditions. That is, models that have complex geometries, straight or curved boundaries, multiple boundaries, transient boundary conditions, multiple aquifer layers, piecewise varying properties and continuously varying properties can be solved. Elements can be implemented using far-field expansions such that model containing many thousands of elements can be solved efficiently to high precision.

The analytic element method has been applied to problems of groundwater flow governed by a variety of linear partial differential equations including the Laplace, the Poisson equation, the modified Helmholtz equation, the heat equation, and the biharmonic equations. Often theses equations are solved using complex variables which enables using mathematical techniques available in complex variable theory. A useful technique to solve complex problems is using conformal mapping which maps the boundary of a geometry, e.g. an ellipse, onto the boundary of the unit circle where the solution is known.

In the analytic element method the discharge potential and stream function, or combined the complex potential, is used. This potential links the physical properties of the groundwater system, the hydraulic head or flow boundaries, to a mathematical representation in a potential. This mathematical representation can be used to calculate the potential in terms of position and thus also solving groundwater flow problems. Elements are developed by solving the boundary conditions for either of these two properties, hydraulic head or flow boundary, which results in analytical solutions capable of dealing with numerous boundary conditions.

A contemporary student of Strack's who is a proponent of the Analytic Element Method (AEM) in groundwater modeling applications is Dr. David Steward of Kansas State University.

## Comparison to other methods

As mentioned the analytic element method thus not rely on discretization of volume or area in the model, as in the finite elements or finite different methods. Thus, it can model complex problem with an error on the order of machine precision. This is illustrated in a study which modeled a highly heterogeneous, isotropic aquifer by including 100,000 spherical heterogeneity with a random conductivity and tracing 40,000 particles.[4] The analytical element method can efficiently be used as verification or as a screening tool in larger projects as it may fast and accurately calculate the groundwater flow for many complex problems.[5][6]

In contrast to other commonly used groundwater modeling methods, e.g. the finite elements or finite different method, the AEM does not discrete the model domain into cells. This gives the advantage that the model is valid for any given point in the model domain. However, it also imposes that the domain is not as easily divided into regions of e.g. different hydraulic conductivity, as when modeling with a cell grid. While there are some solutions that deals with this, e.g. there exist solutions for implementing a vertically varying properties or structures in an aquifer in an AEM model.[7][8][9]

## References

1. ^ Strack, Otto D. L., 1943- (1989). Groundwater mechanics. Englewood Cliffs, N.J.: Prentice Hall. ISBN 0-13-365412-5. OCLC 16276592.CS1 maint: multiple names: authors list (link)
2. ^ Strack, Otto D. L. (August 2017). Analytical Groundwater Mechanics. Cambridge Core. doi:10.1017/9781316563144. ISBN 9781316563144. Retrieved 2020-04-20.
3. ^ Haitjema, H. M. (Henk M.) (1995). Analytic element modeling of groundwater flow. San Diego: Academic Press. ISBN 978-0-08-049910-9. OCLC 162129095.
4. ^ Janković, I.; Fiori, A.; Dagan, G. (2006). "Modeling flow and transport in highly heterogeneous three-dimensional aquifers: Ergodicity, Gaussianity, and anomalous behavior—1. Conceptual issues and numerical simulations". Water Resources Research. 42 (6): W06D12. Bibcode:2006WRR....42.6D12J. doi:10.1029/2005WR004734. ISSN 1944-7973.
5. ^ Hunt, Randall J. (2006). "Ground Water Modeling Applications Using the Analytic Element Method". Groundwater. 44 (1): 5–15. doi:10.1111/j.1745-6584.2005.00143.x. ISSN 1745-6584. PMID 16405461.
6. ^ Kraemer, Stephen R. (2007). "Analytic Element Ground Water Modeling as a Research Program (1980 to 2006)". Groundwater. 45 (4): 402–408. doi:10.1111/j.1745-6584.2007.00314.x. ISSN 1745-6584. PMID 17600570.
7. ^ Bakker, Mark; Strack, Otto D. L. (2003-02-10). "Analytic elements for multiaquifer flow". Journal of Hydrology. 271 (1): 119–129. doi:10.1016/S0022-1694(02)00319-0. ISSN 0022-1694.
8. ^ Strack, O. D. L.; Ausk, B. K. (August 2015). "A formulation for vertically integrated groundwater flow in a stratified coastal aquifer: STRATIFIED COASTAL AQUIFER FLOW". Water Resources Research. 51 (8): 6756–6775. doi:10.1002/2015WR016887.
9. ^ Toller, Erik A. L.; Strack, Otto D. L. (2019). "Interface Flow With Vertically Varying Hydraulic Conductivity". Water Resources Research. 55 (11): 8514–8525. doi:10.1029/2019WR024927. ISSN 1944-7973.