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Splitting solution scheme for material point methodKularathna, Shyamini January 2018 (has links)
Material point method (MPM) is a numerical tool which was originally used for modelling large deformations of solid mechanics problems. Due to the particle based spatial discretiza- tion, MPM is naturally capable of handling large mass movements together with topological changes. Further, the Lagrangian particles in MPM allow an easy implementation of history dependent materials. So far, however, research on MPM has been mostly restricted to explicit dynamic formu- lations with linear approximation functions. This is because of the simplicity and the low computational cost of such explicit algorithms. Particularly in MPM analysis of geomechan- ics problems, a considerable attention is given to the standard explicit formulation to model dynamic large deformations of geomaterials. Nonetheless, several limitations exist. In the limit of incompressibility, a significantly small time step is required to ensure the stability of the explicit formulation. Time step size restriction is also present in low permeability cases in porous media analysis. Spurious pressure oscillations are another numerical instability present in nearly incompressible flow behaviours. This research considers an implicit treatment of the pressure in MPM algorithm to simu- late material incompressibility. The coupled velocity (v)-pressure (p) governing equations are solved by applying Chorin’s projection method which exhibits an inherent pressure stability. Hence, linear finite elements can be used in the MPM solver. The main purpose of this new MPM formulation is to mitigate artificial pressure oscillations and time step restrictions present in the explicit MPM approach. First, a single phase MPM solver is applied to free surface incompressible fluid flow problems. Numerical results show a better approximation of the pressure field compared to the results obtained from the explicit MPM. The proposed formulation is then extended to model fully saturated porous materials with incompress- ible constituents. A solid velocity(v S )-fluid velocity (v F )-pore pressure (p) formulation is presented within the framework of mixture theory. Comparing the numerical results for the one-dimensional consolidation problem shows that the proposed incompressible MPM algorithm provides a stable and accurate pore pressure field even without implementing damping in the solver. Finally, the coupled MPM is used to solve a two-dimensional wave propagation problem and a plain strain consolidation problem. One of the important features of the proposed hydro mechanical coupled MPM formulation is that the time step size is not dependent on the incompressibility and the permeability of the porous medium.
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High order numerical methods for a unified theory of fluid and solid mechanicsChiocchetti, Simone 10 June 2022 (has links)
This dissertation is a contribution to the development of a unified model of
continuum mechanics, describing both fluids and elastic solids as a general
continua, with a simple material parameter choice being the distinction
between inviscid or viscous fluid, or elastic solids or visco-elasto-plastic
media. Additional physical effects such as surface tension, rate-dependent
material failure and fatigue can be, and have been, included in the same
formalism.
The model extends a hyperelastic formulation of solid mechanics in
Eulerian coordinates to fluid flows by means of stiff algebraic relaxation
source terms. The governing equations are then solved by means of high
order ADER Discontinuous Galerkin and Finite Volume schemes on fixed
Cartesian meshes and on moving unstructured polygonal meshes with
adaptive connectivity, the latter constructed and moved by means of a in-
house Fortran library for the generation of high quality Delaunay and Voronoi
meshes.
Further, the thesis introduces a new family of exponential-type and semi-
analytical time-integration methods for the stiff source terms governing
friction and pressure relaxation in Baer-Nunziato compressible multiphase
flows, as well as for relaxation in the unified model of continuum mechanics,
associated with viscosity and plasticity, and heat conduction effects.
Theoretical consideration about the model are also given, from the
solution of weak hyperbolicity issues affecting some special cases of the
governing equations, to the computation of accurate eigenvalue estimates, to
the discussion of the geometrical structure of the equations and involution
constraints of curl type, then enforced both via a GLM curl cleaning method,
and by means of special involution-preserving discrete differential operators,
implemented in a semi-implicit framework.
Concerning applications to real-world problems, this thesis includes
simulation ranging from low-Mach viscous two-phase flow, to shockwaves in
compressible viscous flow on unstructured moving grids, to diffuse interface
crack formation in solids.
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