Bayesian Inverse Problems

Appendices

 

Appendix A: FEM computation of seabed displacements

The governing equilibrium equations to compute the seabed displacements read

$\nabla \cdot \sigma \text{'}-\alpha \nabla p=\rho g+b.$ (A1)

For a saturated porous volume D with surface Σ, the equilibrium is prescribed minimizing the total potential energy by the virtual work theorem. By using the hypothesis of quasi-static conditions and neglecting the inertial forces, the work theorem formulation reads

${\displaystyle {\int }_D\delta \epsilon :}\left(\sigma \text{'}-\alpha p1\right)dD={\displaystyle {\int }_D\delta u\cdot bdD+}{\displaystyle {\int }_{\Gamma N}\delta u\cdot }td\Sigma ,$ (A2)

where ε is the (small) strain tensor, u = [u_x(x̂, ŷ, ẑ) u_y(x̂, ŷ, ẑ) u_z (x̂,ŷ , ẑ)]^T is the displacement vector, 1 the rank-2 identity tensor, and t represents the total external forces acting per unit surface. The symbol δ indicates ...