Solution of problems with material nonlinearities with a coupled finite element/boundary element scheme using an iterative solver. Yucca Mountain Site Characterization Project Page: 15 of 44
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Solution of Nonlinear Cylindrical Cavity Problems Using a
Coupled Finite Element/Boundary Element Technique
This section discusses the solution of cylindrical cavity problems with nonlinear behavior
by using a coupled finite element/boundary element technique. The cavity is surrounded
by a concentric ring of elastic-plastic material, which in turn is surrounded by elastic
material. Since the concentric ring of material around the cavity is a nonlinear material, it
can easily be modeled with finite elements. The surrounding infinite medium of elastic
material is best suited for modeling with boundary elements. This particular problem is
well suited for investigation of the use of a coupling technique for problems involving
nonlinear material behavior. The coupling technique chosen for solution of the problem is
the one described in Reference 9. The finite element method and boundary element method
are coupled by enforcing displacement compatibility and force equilibrium at the nodes
where the finite element and boundary element regions coincide. The resulting set of
equations, which is nonsymmetric, is solved by using a conjugate gradient scheme for
nonsymmetric operators (the Bi-CGSTAB method12). Conjugate gradient schemes have
the advantage that it is not necessary to construct the full stiffness matrix. This eliminates
the problem of large storage requirements for large-scale finite element models. They are
also an efficient solution scheme for large-scale three-dimensional problems.
All of the problems solved in Reference 9 involve completely elastic material. The
problems discussed in this memo examine the coupling scheme when used with problems
exhibiting nonlinear material behavior. The results in the following sections will show that
the coupling scheme is valid for problems involving nonlinear material behavior. For this
class of problems, however, the Bi-CGSTAB method alone is not sufficient as a solution
technique for the resulting set of equations. It becomes necessary to use a solution
approach based on a Newton scheme13. Newton's method is used to set up a linear system
of nonsymmetric equations that are solved with the Bi-CGSTAB method. The reasons for
using this approach are discussed in the following sections.
Because the problems involve nonlinear material, the load must be applied incrementally.
A Newton scheme as outlined in Reference 13 is used to obtain a solution in an incremental
manner. For this process, we define the residual R as the difference between the external
and internal forces for a system. Let AFext represent the increment in the external load for
the current load step, and Fext be the total external load after the load increment AFext is
applied. If the value of the internal forces at the beginning of the load step is Fint(O) , and
the tangent stiffness matrix at the beginning of the load step is KT(o), we can estimate a
displacement increment Au1 corresponding to the current external load increment with the
relation
Aul = K jo)(Fext -Fint(o)), (EQ 18)9
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Koteras, J.R. Solution of problems with material nonlinearities with a coupled finite element/boundary element scheme using an iterative solver. Yucca Mountain Site Characterization Project, report, January 1, 1996; Albuquerque, New Mexico. (https://digital.library.unt.edu/ark:/67531/metadc667311/m1/15/: accessed July 16, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.