Computational stress analysis using finite volume methods

There is a growing interest in applying finite volume methods to model solid mechanics problems and multi-physics phenomena. During the last ten years an increasing amount of activity has taken place in this area. Unlike the finite element formulation, which generally involves volume integrals, the finite volume formulation transfers volume integrals to surface integrals using the divergence theorem. This transformation for convection and diffusion terms in the governing equations, ensures conservation at the local element level. This is seen as a major attraction for finite volume methods. The research presented in this thesis details the development of a cell vertex based finite volume formulation for complex analysis like geometrically nonlinear modelling and plate analysis. For both geometrically nonlinear and plate analysis a series of simulation results are presented and are compared with conventional finite element results. Further research has been carried out to solve stress problems in multi-physics phenomena using a Computational Fluid Dynamics(CFD) frame-work. This approach has the advantage in that it uses the similarities between fluid and solid momentum equations to introduce some modifications in a CFD code that allows a complete CFD solution procedure to be used for the simultaneous calculation of the velocity, temperature and displacement variables. The results of this integrated approach are compared with results obtained by using techniques which solve the problem by 111 using two solvers (one for solid regions, one for fluid regions). In summary, the novelty of the research detailed in this thesis is: • Finite volulne formulation for elastic large strain analysis. Comparison of this approach with traditional finite element techniques: - Cell-vertex finite volume method is as accurate as finite element approach but slower in solution time. • Finite volume formulation for structural plate analysis. Comparison with traditional finite element method: - Novel finite volume approach is as accurate as finite element approach. - Does not display locking problems (observed with finite element methods) and is comparable in solution times. • Formulation of an integrated CFD solver for coupled flow, heat transfer and stress calculations. Comparison with a 2-solver approach: - Integrated approach is much faster and substantially less memory intensive than 2-solver approach. Comparisons between the new formulation and traditional approaches are made in terms of accuracy and solution speed

Adaptive vertex-centered finite volume methods for general second-order linear elliptic PDEs

We prove optimal convergence rates for the discretization of a general second-order linear elliptic PDE with an adaptive vertex-centered finite volume scheme. While our prior work Erath and Praetorius [SIAM J. Numer. Anal., 54 (2016), pp. 2228--2255] was restricted to symmetric problems, the present analysis also covers non-symmetric problems and hence the important case of present convection

Finite volume schemes for diffusion equations: introduction to and review of modern methods

We present Finite Volume methods for diffusion equations on generic meshes, that received important coverage in the last decade or so. After introducing the main ideas and construction principles of the methods, we review some literature results, focusing on two important properties of schemes (discrete versions of well-known properties of the continuous equation): coercivity and minimum-maximum principles. Coercivity ensures the stability of the method as well as its convergence under assumptions compatible with real-world applications, whereas minimum-maximum principles are crucial in case of strong anisotropy to obtain physically meaningful approximate solutions

Superconvergence Using Pointwise Interpolation in Convection-Diffusion Problems

Considering a singularly perturbed convection-diffusion problem, we present an analysis for a superconvergence result using pointwise interpolation of Gau{\ss}-Lobatto type for higher-order streamline diffusion FEM. We show a useful connection between two different types of interpolation, namely a vertex-edge-cell interpolant and a pointwise interpolant. Moreover, different postprocessing operators are analysed and applied to model problems.Comment: 19 page

VAGO method for the solution of elliptic second-order boundary value problems

Mathematical physics problems are often formulated using differential oprators of vector analysis - invariant operators of first order, namely, divergence, gradient and rotor operators. In approximate solution of such problems it is natural to employ similar operator formulations for grid problems, too. The VAGO (Vector Analysis Grid Operators) method is based on such a methodology. In this paper the vector analysis difference operators are constructed using the Delaunay triangulation and the Voronoi diagrams. Further the VAGO method is used to solve approximately boundary value problems for the general elliptic equation of second order. In the convection-diffusion-reaction equation the diffusion coefficient is a symmetric tensor of second order

A New Domain Decomposition Method for the Compressible Euler Equations

In this work we design a new domain decomposition method for the Euler equations in 2 dimensions. The basis is the equivalence via the Smith factorization with a third order scalar equation to whom we can apply an algorithm inspired from the Robin-Robin preconditioner for the convection-diffusion equation. Afterwards we translate it into an algorithm for the initial system and prove that at the continuous level and for a decomposition into 2 sub-domains, it converges in 2 iterations. This property cannot be preserved strictly at discrete level and for arbitrary domain decompositions but we still have numerical results which confirm a very good stability with respect to the various parameters of the problem (mesh size, Mach number, ....).Comment: Submitte

A collocated finite volume scheme to solve free convection for general non-conforming grids

We present a new collocated numerical scheme for the approximation of the Navier-Stokes and energy equations under the Boussinesq assumption for general grids, using the velocity-pressure unknowns. This scheme is based on a recent scheme for the diffusion terms. Stability properties are drawn from particular choices for the pressure gradient and the non-linear terms. Numerical results show the accuracy of the scheme on irregular grids