Finite Element Methods in Soil Mechanics

Transcription

1 Aalto University Department of Mathematics and Systems Analysis Graduate School in Engineering Mechanics Annual Seminar

2 Background Master's Thesis late 2007 Topic: composite Reissner-Mindlin plates Mechanics major Mechanics of Materials minor Ever since: PhD studies with Rolf Stenberg Finite elements in soil mechanics Past two years on a grant from the Finnisch cultural foundation

3 Soil Mechanics We have primarily focused on the following two problems: Biot consolidation model Consolidation of soil, e.g. sand, clay, and peat Time-dependent Numerical challenges: stability, fast solvers Brinkman model Flow in very porous medium Parameter-dependent Numerical challenges: robustness, error control, large systems

4 Biot model What is being modelled? Movements of saturated soil Long time scales Typical applications Oil well deformations, nuclear waste disposal Enbankments, foundations, and roads on soft soil Practical problem: how large should the concrete footing pads be made?

5 Biot model Mixed problem: the displacement u and the pore pressure p Stability issues Correct FE-space combination for u and p Boundary layers near t = 0 Parabolic equation: time-integration Our approach: Present a simple mathematical analysis Use well-established Stokes elements Keep system complexity low Co-operation with KYT2010 and Numerola Oy

6 Brinkman model What is being modelled? Fluid ow in porous medium Parametrized problem that covers both of the limiting cases: Darcy and Stokes ow Typical applications Oil reservoirs, groundwater modelling Transition layer between free and porous ow Special applications: heat pipes, ltration Practical problem: how to keep the little fun things in life?

7 Brinkman model Mixed problem: the velocity u and the pressure p Parametrized by eective viscosity t At t = 0, we get the Darcy ow With t large a Stokes-type ow Robustness: method that works for all values of t Error control: mesh renements, reliable upper bounds Large systems Real-life systems in 3D million unknowns M.F.Wheeler: "No-one gives a damn about Poisson equation in a unit square" Iterative solvers eective preconditioners crucial Memory issues, system complexity

8 Status of the thesis Two papers submitted in 2009: Analysis of H(div)-conforming nite elements for the Brinkman problem. (JK & Rolf Stenberg) Thorough theoretical analysis, including a posteriori estimators. Submitted to Mathematical Models and Methods in Applied Sciences Mixed Finite Element Methods for Problems with Robin Boundary Conditions (JK, RS & Dominik Schötzau) Deals with setting general boundary conditions easily for the Darcy ow Submitted to SIAM Journal on Numerical analysis One paper published: Based on M.Sc. work. Published in Finite Elements in Analysis & Design

9 Ongoing work Articles in preparation: Theoretical analysis of the Biot equations, ready Feb-Mar 2010 Numerical results for the Brinkman model, ready summer 2010 Research tasks planned for 2010: Continue development of FE solver for the Brinkman equations: matrix free implementation, 3D elements Preconditioner for the Brinkman equations. Joint work with Antti Hannukainen. Numerical experiments for the Biot model. Collaboration with Numerola Oy, real-life geometry for the nuclear waste disposal capsules.

10 Open questions Biot model Suitable combination of timestepping and nite elements? How to keep solving the FE-problem cheap? Adaptivity? Brinkman model We have veried the behavior predicted theoretically. How can we handle the jump from t = 0? What is an ecient preconditioner for the linear system? Do Darcy elements give an advantage over Stokes elements?

11 Timetable Before Fall 2010 Finalize the Biot article. Numerical results for Brinkman, write article. Fingers crossed for the two submissions to be accepted. Second article on Biot model. Maybe look into Brinkman with anisotropy? Thesis ready mid-2011