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The Implicit Material Point Method for Variable Viscosity Stokes Flow

David Alexander May

出版	Monash University, 2009
URL	http://books.google.com.hk/books?id=j09WMwEACAAJ&hl=&source=gbs_api

註釋

The outline of this thesis is as follows: In Chapter 1 we provide a general overview of the types of geodynamic processes we are interesting in modelling. The incompressible Stokes flow equations are introduced and serve as our underlying continuum description of the Earth's mantle, lithosphere and crust over long time scales. We close the chapter by describing various numerical modelling techniques which are suitable for studying problems in geodynamics. The implicit material point method (IMPM) is described in Chapter 2. Here we analyse the variable coefficient Poisson equation. We focus on understanding the convergence behaviour of IMPM when the PDE contains smoothly varying coefficients. Estimates of the order of the quadrature rule required to evalulate the variational problem are derived. The case of discontinuous PDE coefficients is also discussed. In Chapter 3 we apply the IMPM discretisation to incompressible Stokes flow. The theory from the proceeding chapter is used to develop a low order method suited to modelling variable viscosity Stokes flow. The construction of the material point basis functions in multiple spatial dimensions is described, together with quadrature techniques to evaluate the discrete variational problem. The convergence of IMPM when applied to Stokes flow with both smoothly varying and discontinuous viscosity fields is examined numerically. We establish that the numerical experiments support the theoretical expectations. The flexibility of the IMPM discretisation is highlighted by demonstrating how it may be coupled to an interface tracking scheme such as the level set method. Within Chapter 4, we describe a technique to improve the estimates of total stress, deviatoric stress, strain rate and pressure which are obtained from both classical finite elements and IMPM. The approach used is known as the recovery by equilibrium of patches (REP) and its integral formulation is amenable to any method which poses the continuum equation as a variational problem. We demonstrate that REP can be used to generate super-convergent values for the total stress, deviatoric stress and strain rate using either traditional finite elements or IMPM for Stokes flow problems with smoothly varying viscosity fields. In addition, we demonstrate a technique to compute super-convergent pressure fields. We discuss the issues and propose a method to use REP with IMPM when the viscosity field is discontinuous.