244 research outputs found
Almost- splines: Biquadratic splines on unstructured quadrilateral meshes and their application to fourth order problems
Isogeometric Analysis generalizes classical finite element analysis and
intends to integrate it with the field of Computer-Aided Design. A central
problem in achieving this objective is the reconstruction of analysis-suitable
models from Computer-Aided Design models, which is in general a non-trivial and
time-consuming task. In this article, we present a novel spline construction,
that enables model reconstruction as well as simulation of high-order PDEs on
the reconstructed models. The proposed almost- are biquadratic splines on
fully unstructured quadrilateral meshes (without restrictions on placements or
number of extraordinary vertices). They are smooth almost everywhere,
that is, at all vertices and across most edges, and in addition almost (i.e.
approximately) smooth across all other edges. Thus, the splines form
-nonconforming analysis-suitable discretization spaces. This is the
lowest-degree unstructured spline construction that can be used to solve
fourth-order problems. The associated spline basis is non-singular and has
several B-spline-like properties (e.g., partition of unity, non-negativity,
local support), the almost- splines are described in an explicit
B\'ezier-extraction-based framework that can be easily implemented. Numerical
tests suggest that the basis is well-conditioned and exhibits optimal
approximation behavior
Mathematical and computational aspects of the enhanced strain finite element method
Bibliography: pages 102-107.This thesis deals with further investigations of the enhanced strain finite element method, with particular attention given to the analysis of the method for isoparametric elements. It is shown that the results established earlier by B D Reddy and J C Simo for affine-equivalent meshes carry over to the case of isoparameric elements. That is, the method is stable and convergent provided that a set of three conditions are met, and convergence is at the same rate as in the standard method. The three conditions differ in some respects, though, from their counterparts for the affine case. A procedure for recovering the stress is shown to lead to an approximate stress which converges at the optimal rate to the actual stress. The concept of the equivalent parallelogram associated with a quadrilateral is introduced. The quadrilateral may be regarded as a perturbation of this parallelogram, which is most conveniently described by making use of properties of the isoparametric map which defines the quadrilateral. The equivalent parallelogram generates a natural means of defining a regular family of quadrilaterals; this definition is used together with other properties to obtain in a relatively simple manner estimates, in appropriate seminorms or norms, of the isoparametric map and it's Jacobian, for use in the determination of finite element interpolation error estimates, with regard to computations, a new basis for enhanced strains is introduced, and various examples have been tested. The results obtained are compared with those obtained using other bases, and with those found from an assumed stress approach. Favourable comparisons are obtained in most cases, with the present basis exhibiting an improvement over existing bases. Convergence of the finite element results are verified; it is observed numerically that the improvement of results due to enhancement is as a result of a smaller constant appearing in the error estimates
Finite element analysis of slope stability
Slope stability is an important aspect of geotechnical engineering. The use of finite element analysis of slope stability has gained popularity in recent years due to its capability to handle complex problems. The primary focus of this research was to study the influence of soil nailing on the factor of safety of stability of slopes by using finite element analysis, and to investigate failure mechanism. In this paper, stability of various soil slopes was analyzed using the Shear Strength Reduction (SSR) technique. Finite element analysis was performed on both reinforced and unreinforced soil slopes and the results were compared with results from various traditional methods. Finite element results show that analysis of stability of slopes using the SSR technique is a useful alternative compared to traditional methods, especially when geometry is complex
Efficient and robust monolithic finite element multilevel Krylov subspace solvers for the solution of stationary incompressible Navier-Stokes equations
Multigrid methods belong to the best-known methods for solving linear systems
arising from the discretization of elliptic partial differential equations. The
main attraction of multigrid methods is that they have an asymptotically meshindependent
convergence behavior. Multigrid with Vanka (or local multilevel
pressure Schur complement method) as smoother have been frequently used for
the construction of very effcient coupled monolithic solvers for the solution of
the stationary incompressible Navier-Stokes equations in 2D and 3D. However,
due to its innate Gauß-Seidel/Jacobi character, Vanka has a strong influence
of the underlying mesh, and therefore, coupled multigrid solvers with Vanka
smoothing very frequently face convergence issues on meshes with high aspect
ratios. Moreover, even on very nice regular grids, these solvers may fail when
the anisotropies are introduced from the differential operator.
In this thesis, we develop a new class of robust and efficient monolithic finite
element multilevel Krylov subspace methods (MLKM) for the solution of the
stationary incompressible Navier-Stokes equations as an alternative to the coupled
multigrid-based solvers. Different from multigrid, the MLKM utilizes a
Krylov method as the basis in the error reduction process. The solver is based
on the multilevel projection-based method of Erlangga and Nabben, which accelerates
the convergence of the Krylov subspace methods by shifting the small
eigenvalues of the system matrix, responsible for the slow convergence of the
Krylov iteration, to the largest eigenvalue.
Before embarking on the Navier-Stokes equations, we first test our implementation
of the MLKM solver by solving scalar model problems, namely the
convection-diffusion problem and the anisotropic diffusion problem. We validate
the method by solving several standard benchmark problems. Next, we
present the numerical results for the solution of the incompressible Navier-Stokes
equations in two dimensions. The results show that the MLKM solvers produce
asymptotically mesh-size independent, as well as Reynolds number independent
convergence rates, for a moderate range of Reynolds numbers. Moreover, numerical
simulations also show that the coupled MLKM solvers can handle (both
mesh and operator based) anisotropies better than the coupled multigrid solvers
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