Finite Element Analysis of Lithospheric Deformation
Victor M. Calo,
(King Abdullah University of Sciences and Technology KAUST, Saudi Arabia)
Jueves 22 de noviembre 2012
To address the challenging lithospheric deformation problems, we have proposed two alternative approaches. The first one uses an Eulerian formulation to describe the elastic deformation of a solid. The second one developed efficient and flexible unstructured FEM code to study lithospheric scale deformation. First, we will present a numerical formulation aimed at modeling the nonlinear response of elastic materials using large deformation continuum mechanics in three dimensions. This finite element formulation is based on the Eulerian description of motion and the transport of the deformation gradient. When modeling a nearly incompressible solid, the transport of the deformation gradient is decomposed into its isochoric part and the Jacobian determinant as independent fields. A homogeneous isotropic hyperelastic solid is assumed and B-splines-based finite elements are used for the spatial discretization. A variational multiscale residual-based approach is employed to stabilize the transport equations. The performance of the scheme is explored for both compressible and nearly incompressible applications. The numerical results are in good agreement with theory illustrating the viability of the computational scheme. Secondly, we will describe efficient and flexible unstructured finite element discretization that is superior to traditional ones. That is, linear elements on a simplex has serious volumetric locking problems for quasi-incompressible material. The additional incompressibility constraint makes the numerical problem over-determined, which results in inaccurate solutions characterized by spurious oscillations.
Recently, Micheli and Morcellin (2009) proposed a novel and efficient anti-volumetric locking method based on traditional linear finite element method. The essence of their method lies on calculating the volumetric strain from the actual volume change of the element instead of from strain rate accumulation. We adapted the original finite-difference FLAC (Fast Lagrangian Analysis of Continua) algorithm to finite element formulation with the anti-volumetric locking modification. The new method has several advantages over the FLAC method. First, the mixed discretization and overlapping tetrahedra are no longer necessary, reducing the arithmetic operations by 50%. Second, finite element formulation allows simpler integration of the stresses and velocities, reducing the arithmetic operations further by 10%. Third, switching from a regular rectangular mesh to an unstructured tetrahedral mesh gives flexibility and improves the accuracy in tracking of internal boundaries, such as the Moho discontinuity, fault zones, and material interfaces. We will demonstrate the capability of the code in two cases: (1) reproducing the thermo-chemical convection benchmark (van Keken et al., 1997) to validate the algorithm with pure viscous rheology; (2) modeling spontaneous formation of normal faults by the reduction of cohesion in a frictional and cohesive elasto-plastic layer.