Generic weak form (single field)

Ratel solves the momentum balance equations using unstructured high-order finite/spectral element spatial discretizations by matrix-free approach. We present here the notation and mathematical formulation of matrix-free method for a general Dirichlet problem \(\bm{R}(\bm{u}) = 0\): find \(\bm{u} \in \mathcal{V} \subset H^1 \left( \Omega_0 \right)\) such that

(138)\[ \langle\bm{v}, \bm{R}(\bm{u}) \rangle= \int_{\Omega_0}{\bm{v} \cdot \bm{f}_0 (\bm{u}, \nabla \bm{u}) + \nabla \bm{v} \tcolon \bm{f}_1 (\bm{u}, \nabla \bm{u})} \, dV = 0, \quad \forall \bm{v} \in \mathcal{V}, \]

where the operators \(\bm{f}_0\) and \(\bm{f}_1\) contain all possible sources in the problem. In order to solve (138) with Newton-Krylov iterative solvers we need its Jacobian form: find \(\diff \bm{u} \in \mathcal{V} \subset H^1 \left( \Omega_0 \right)\) such that

(139)\[ \langle\bm{v}, \bm{J}(\bm{u}) \diff \bm{u} \rangle= \int_{\Omega_0}{\bm{v} \cdot \diff \bm{f}_0 + \nabla \bm{v} \tcolon \diff \bm{f}_1} \, dV, \]

where the linearization of operators \(\bm{f}_0\) and \(\bm{f}_1\) are

\[ \diff \bm{f}_i = \frac{\partial \bm{f}_i}{\partial \bm{u}} \diff \bm{u} + \frac{\partial \bm{f}_i}{\partial \nabla \bm{u}} \nabla \diff \bm{u}, \quad i=0,1 \]

It should be noted that the gradient in the (138), (139) depends on the configuration system and it could be with respect to initial configuration \(\bm{X}\) i.e., \(\nabla_X \bm{v}\) (Lagrangian approach) or current configuration \(\bm{x}\) i.e., \(\nabla_x \bm{v}\) (Eulerian approach).

Compare with governing equations derived in pervious sections for linear and large deformation, it is easy to verify that

  • For linear elasticity described in (82) we have

\[ \bm f_0 = -\rho \bm g, \quad \bm f_1 = \bm \sigma(\bm u). \]

where the linearization satisfies \(\diff\bm f_{1}(\diff\bm u) = \bm f_1(\diff \bm u)\) due to linearity.

  • For hyperelastic in initial configuration described in (93) and (95) we have

\[ \bm f_0 = -\rho_0 \bm g, \quad \bm f_1 = \bm F \bm S, \quad \diff \bm f_1 = \diff \bm F \bm S + \bm F \diff \bm S. \]

  • For hyperelastic in current configuration derived in (103) and (106)

\[ \bm f_0 = -\rho_0 \bm g, \quad \bm f_1 = \bm \tau, \quad \diff \bm f_1 = \diff \bm{\tau} - \bm{\tau} \left( \nabla_x \diff \bm{u} \right)^T. \]