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Title: Application of Mass Lumped Higher Order Finite Elements

Abstract

There are many interesting phenomena in extended-MHD such as anisotropic transport, mhd, 2-fluid effects stellarator and hot particles. Any one of them challenges numerical analysts, and researchers are seeking for higher order methods, such as higher order finite difference, higher order finite elements and hp/spectral elements. It is true that these methods give more accurate solution than their linear counterparts. However, numerically they are prohibitively expensive. Here we give a successful solution of this conflict by applying mass lumped higher order finite elements. This type of elements not only keep second/third order accuracy but also scale closely to linear elements by doing mass lumping. This is especially true for second order lump elements. Full M3D and anisotropic transport models are studied.

Authors:
; ; ; ; ; ;
Publication Date:
Research Org.:
Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States)
Sponsoring Org.:
USDOE Office of Science (SC)
OSTI Identifier:
934516
Report Number(s):
PPPL-4128
TRN: US0804644
DOE Contract Number:
AC02-76CH03073
Resource Type:
Technical Report
Country of Publication:
United States
Language:
English
Subject:
70 PLASMA PHYSICS AND FUSION TECHNOLOGY; ACCURACY; STELLARATORS; TRANSPORT; Higher order finite elements; mass lumping; highly anisotropic transport

Citation Formats

Chen, J., Strauss, H. R., Jardin, S. C., Park, W., Sugiyama, L. E., G. Fu, and Breslau, J. Application of Mass Lumped Higher Order Finite Elements. United States: N. p., 2005. Web. doi:10.2172/934516.
Chen, J., Strauss, H. R., Jardin, S. C., Park, W., Sugiyama, L. E., G. Fu, & Breslau, J. Application of Mass Lumped Higher Order Finite Elements. United States. doi:10.2172/934516.
Chen, J., Strauss, H. R., Jardin, S. C., Park, W., Sugiyama, L. E., G. Fu, and Breslau, J. Tue . "Application of Mass Lumped Higher Order Finite Elements". United States. doi:10.2172/934516. https://www.osti.gov/servlets/purl/934516.
@article{osti_934516,
title = {Application of Mass Lumped Higher Order Finite Elements},
author = {Chen, J. and Strauss, H. R. and Jardin, S. C. and Park, W. and Sugiyama, L. E. and G. Fu and Breslau, J.},
abstractNote = {There are many interesting phenomena in extended-MHD such as anisotropic transport, mhd, 2-fluid effects stellarator and hot particles. Any one of them challenges numerical analysts, and researchers are seeking for higher order methods, such as higher order finite difference, higher order finite elements and hp/spectral elements. It is true that these methods give more accurate solution than their linear counterparts. However, numerically they are prohibitively expensive. Here we give a successful solution of this conflict by applying mass lumped higher order finite elements. This type of elements not only keep second/third order accuracy but also scale closely to linear elements by doing mass lumping. This is especially true for second order lump elements. Full M3D and anisotropic transport models are studied.},
doi = {10.2172/934516},
journal = {},
number = ,
volume = ,
place = {United States},
year = {Tue Nov 01 00:00:00 EST 2005},
month = {Tue Nov 01 00:00:00 EST 2005}
}

Technical Report:

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  • The M3D code has been using linear finite elements to represent multilevel MHD on 2-D poloidal planes. Triangular higher order elements, up to third order, are constructed here in order to provide M3D the capability to solve highly anisotropic transport problems. It is found that higher order elements are essential to resolve the thin transition layer characteristic of the anisotropic transport equation, particularly when the strong anisotropic direction is not aligned with one of the Cartesian coordinates. The transition layer is measured by the profile width, which is zero for infinite anisotropy. It is shown that only higher order schemesmore » have the ability to make this layer converge towards zero when the anisotropy gets stronger and stronger. Two cases are considered. One has the strong transport direction partially aligned with one of the element edges, the other doesn't have any alignment. Both cases have the strong transport direction misaligned with the grid line by some angles.« less
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  • Finite element meshes are used to approximate the solution to some differential equation when no exact solution exists. A finite element mesh consists of many small (but finite, not infinitesimal or differential) regions of space that partition the problem domain, {Omega}. Each region, or element, or cell has an associated polynomial map, {Phi}, that converts the coordinates of any point, x = ( x y z ), in the element into another value, f(x), that is an approximate solution to the differential equation, as in Figure 1(a). This representation works quite well for axis-aligned regions of space, but when theremore » are curved boundaries on the problem domain, {Omega}, it becomes algorithmically much more difficult to define {Phi} in terms of x. Rather, we define an archetypal element in a new coordinate space, r = ( r s t ), which has a simple, axis-aligned boundary (see Figure 1(b)) and place two maps onto our archetypal element:« less
  • This report contains an algorithm for decomposing higher-order finite elementsinto regions appropriate for isosurfacing and proves the conditions under which thealgorithm will terminate. Finite elements are used to create piecewise polynomialapproximants to the solution of partial differential equations for which no analyticalsolution exists. These polynomials represent fields such as pressure, stress, and mo-mentim. In the past, these polynomials have been linear in each parametric coordinate.Each polynomial coefficient must be uniquely determined by a simulation, and thesecoefficients are called degrees of freedom. When there are not enough degrees of free-dom, simulations will typically fail to produce a valid approximation to themore » solution.Recent work has shown that increasing the number of degrees of freedom by increas-ing the order of the polynomial approximation (instead of increasing the number offinite elements, each of which has its own set of coefficients) can allow some typesof simulations to produce a valid approximation with many fewer degrees of freedomthan increasing the number of finite elements alone. However, once the simulation hasdetermined the values of all the coefficients in a higher-order approximant, tools donot exist for visual inspection of the solution.This report focuses on a technique for the visual inspection of higher-order finiteelement simulation results based on decomposing each finite element into simplicialregions where existing visualization algorithms such as isosurfacing will work. Therequirements of the isosurfacing algorithm are enumerated and related to the placeswhere the partial derivatives of the polynomial become zero. The original isosurfacingalgorithm is then applied to each of these regions in turn.3 AcknowledgementThe authors would like to thank David Day and Louis Romero for their insight into poly-nomial system solvers and the LDRD Senior Council for the opportunity to pursue thisresearch. The authors were supported by the United States Department of Energy, Officeof Defense Programs by the Labratory Directed Research and Development Senior Coun-cil, project 90499. Sandia is a multiprogram laboratory operated by Sandia Corporation,a Lockheed-Martin Company, for the United States Department of Energy under contractDE-AC04-94-AL85000.4« less
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