Download Finite Element Analysis of Lithospheric Deformation Victor M. Calo

Finite Element Analysis of Lithospheric Deformation
Victor M. Calo, Eunseo Choi, Nathan O. Collier, Ravindra Duddu, Eh Tan, Luc L. Lavier
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-splinesbased finite elements are used for the spatial discretization. A variational multiscale residualbased 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 (A new efficient explicit formulation for
linear tetrahedral elements non-sensitive to volumetric locking for infinitesimal elasticity and
inelasticity. Int. J. Numer. Meth. Engng., v. 79, pp. 45-68, 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 cohesional elasto-plastic layer.