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Title: Finite Dimensional Approximations for Continuum Multiscale Problems

Abstract

The completed research project concerns the development of novel computational techniques for modeling nonlinear multiscale physical and biological phenomena. Specifically, it addresses the theoretical development and applications of the homogenization theory (coarse graining) approach to calculation of the effective properties of highly heterogenous biological and bio-inspired materials with many spatial scales and nonlinear behavior. This theory studies properties of strongly heterogeneous media in problems arising in materials science, geoscience, biology, etc. Modeling of such media raises fundamental mathematical questions, primarily in partial differential equations (PDEs) and calculus of variations, the subject of the PI’s research. The focus of completed research was on mathematical models of biological and bio-inspired materials with the common theme of multiscale analysis and coarse grain computational techniques. Biological and bio-inspired materials offer the unique ability to create environmentally clean functional materials used for energy conversion and storage. These materials are intrinsically complex, with hierarchical organization occurring on many nested length and time scales. The potential to rationally design and tailor the properties of these materials for broad energy applications has been hampered by the lack of computational techniques, which are able to bridge from the molecular to the macroscopic scale. The project addressed the challenge ofmore » computational treatments of such complex materials by the development of a synergistic approach that combines innovative multiscale modeling/analysis techniques with high performance computing.« less

Authors:
 [1]
  1. Pennsylvania State Univ., University Park, PA (United States)
Publication Date:
Research Org.:
Pennsylvania State Univ., University Park, PA (United States)
Sponsoring Org.:
USDOE Office of Science (SC)
OSTI Identifier:
1340478
Report Number(s):
DOE-PSU-25862
DOE Contract Number:
FG02-08ER25862
Resource Type:
Technical Report
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; bio-mimetic; multiscale modeling; coarse-graining; computational complexity; high performance computing

Citation Formats

Berlyand, Leonid. Finite Dimensional Approximations for Continuum Multiscale Problems. United States: N. p., 2017. Web. doi:10.2172/1340478.
Berlyand, Leonid. Finite Dimensional Approximations for Continuum Multiscale Problems. United States. doi:10.2172/1340478.
Berlyand, Leonid. Tue . "Finite Dimensional Approximations for Continuum Multiscale Problems". United States. doi:10.2172/1340478. https://www.osti.gov/servlets/purl/1340478.
@article{osti_1340478,
title = {Finite Dimensional Approximations for Continuum Multiscale Problems},
author = {Berlyand, Leonid},
abstractNote = {The completed research project concerns the development of novel computational techniques for modeling nonlinear multiscale physical and biological phenomena. Specifically, it addresses the theoretical development and applications of the homogenization theory (coarse graining) approach to calculation of the effective properties of highly heterogenous biological and bio-inspired materials with many spatial scales and nonlinear behavior. This theory studies properties of strongly heterogeneous media in problems arising in materials science, geoscience, biology, etc. Modeling of such media raises fundamental mathematical questions, primarily in partial differential equations (PDEs) and calculus of variations, the subject of the PI’s research. The focus of completed research was on mathematical models of biological and bio-inspired materials with the common theme of multiscale analysis and coarse grain computational techniques. Biological and bio-inspired materials offer the unique ability to create environmentally clean functional materials used for energy conversion and storage. These materials are intrinsically complex, with hierarchical organization occurring on many nested length and time scales. The potential to rationally design and tailor the properties of these materials for broad energy applications has been hampered by the lack of computational techniques, which are able to bridge from the molecular to the macroscopic scale. The project addressed the challenge of computational treatments of such complex materials by the development of a synergistic approach that combines innovative multiscale modeling/analysis techniques with high performance computing.},
doi = {10.2172/1340478},
journal = {},
number = ,
volume = ,
place = {United States},
year = {Tue Jan 24 00:00:00 EST 2017},
month = {Tue Jan 24 00:00:00 EST 2017}
}

Technical Report:

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  • A new class of hybrid finite element methods for the numerical analysis of second-order elliptic boundary-value problems is presented. The methods are characterized by the use of particular solutions of the differential equation being solved, in contrast to conventional hybrid methods, in which polynomial approximations are used. As a model problem the Dirichlet problem for Laplace's equation is studied. A priori error estimates are derived, and the results of numerical experiments are presented. 3 figures, 1 table.
  • Mixed-hybrid finite element approximations are described for second-order elliptic boundary-value problems in which independent approximations are used for the solution and its gradient on the interior of an element and the trace of the gradients on the boundary of the element. This lead to nonconforming finite elements. The independent boundary approximations are introduced by means of Lagrange multipliers. Error estimates are derived a priori. Several other finite element models are also obtained as special cases. (auth)
  • The AFTON code calculates the motion of a bounded continuum, or collection of continua, in an arbitrary, time-dependent coordinate system. The complete set of finite-difference equations for the revised and improved version is presented, as well as details of the revisions and a partial bibliography of applications.
  • Fd2 is a software package developed at Teledyne Geotech Alexandria Laboratories (TGAL) during the past several years for generating synthetic seismograms and displaying the wavefields. This package consists of primarily a 2-dimensional 2nd-order explicit linear finite-difference (LFD) code. LFD method has the advantage that the solution contains all conversions and all orders of multiple scattering. It permits examinations of fairly general models with arbitrary complex variations in material properties and free-surface geometry. Furthermore, it does not require many assumptions commonly invoked in other theoretical approaches. The basic limitations to the LFD method or the finite-element method are the computational costmore » and memory requirements. These constrain the size of the grid and the number of time steps that can be calculated over a reasonable time frame. Our LFD code has a distinguishable feature in that it allows the inclusion o topographical free surface. This is particularly useful in modeling nuclear explosions buried in mountains. In this topical report, sample scripts are presented to illustrate the usage of fd2 and several supporting routines for plotting out the synthetics, generating 2-dimensional media, as well as the graphic visualization of wavefields. The algorithms for handling the boundary conditions of polygonal topography are reviewed in detail. Thus this topical report serves as both a programmer's guide and the user's manual.« less
  • The computer program HT2D for performing finite element, two-dimensional, conduction heat transfer analyses is presented. The program can be used on UNIVAC or CYBER computers. The basic element used is the isoparametric quadrilateral in either cartesian coordinates or cylindrical coordinates. Conduction and convection coefficients are assumed to be constant in time but not space. Internal heat generation is included and may be varied from element to element. The program has the unusual feature in that a backward difference implicit integration scheme is used in time. The user needs to supply only initial temperatures to restart the program. The program onmore » the CYBER uses the variable dimension concept. A complete derivation is included. (LCL)« less