# Mixed Method for Isogeometric Analysis of Coupled Flow and Deformation in Poroelastic Media
## 2 Governing Equations
### Galerkin Finite Element Formulation  
With a suitable number \(N\) of basis functions defined, let \(\mathbf{N}_{\mathrm{p}}:\Omega\to\mathbb{R}^{1\times N}\) and \(\mathbf{N}_{\mathrm{u}}:\Omega\to\mathbb{R}^{d\times dN}\) be the basis interpolation matrices for the pressure and displacement, respectively. The unknowns and the test functions can then be represented using coefficient vectors:  
\[ \begin{split}\mathbf{u}=\mathbf{N}_{\mathrm{u}}\mathbf{u}^{\mathrm{c}},& \delta\mathbf{u}=\mathbf{N}_{\mathrm{u}}\delta\mathbf{u}^{\mathrm{c}},\\ p^{\mathrm{f}}=\mathbf{N}_{\mathrm{p}}\mathbf{p}^{\mathrm{c}},& \delta p=\mathbf{N}_{\mathrm{p}}\delta p^{\mathrm{c}}\end{split}  \tag{26}\]  
where \(\mathbf{u}^{\mathrm{c}}\) and \(\mathbf{p}^{\mathrm{c}}\) are the control point values of the displacement and pressure field variables. Application of (26) to the weak form of the linear momentum balance equation in (21) results in the matrix the discrete system of equations (after canceling \(\delta\mathbf{u}^{\mathrm{c}}\) and \(\delta p^{\mathrm{c}}\), as Equations (7) and (14) are supposed to be valid for any choice of these)  
\[\mathbf{K}\mathbf{u}^{\mathrm{c}}-\mathbf{Q}\mathbf{p}^{\mathrm{c}}=\mathbf{f}_{\mathrm{u}} \tag{27}\]  
where the stiffness matrix \(\mathbf{K}\), the coupling matrix \(\mathbf{Q}\) and the vector of body forces and surface tractions \(\mathbf{f}_{u}\) are given by  
\[ \begin{split}\mathbf{K}&=\int_{\Omega}\mathbf{B}^{\mathrm{ T}}\tilde{\mathbf{D}}\mathbf{B}\mathrm{d}\Omega,\\ \mathbf{Q}&=\int_{\Omega}\mathbf{B}^{\mathrm{T}}\alpha\tilde{ \mathbf{I}}\mathbf{N}_{\mathrm{p}}\mathrm{d}\Omega\\ f_{\mathrm{u}}&=\int_{\Omega}\mathbf{N}_{\mathrm{u}}^{ \mathrm{T}}\rho\mathbf{b}\mathrm{d}\Omega+\int_{\Gamma_{N}^{\mathrm{a}}}\mathbf{N}_{ \mathrm{u}}^{\mathrm{T}}\mathbf{\mathrm{f}}\mathrm{d}\Gamma.\end{split}  \tag{28}\]  
Here \(\mathbf{B}=\mathbf{L}\mathbf{N}_{\mathrm{u}}\) is the strain-displacement matrix. Similarly, using (26) in the weak form of the mass balance equation in (25) results in the discrete system of equations  
\[\mathbf{Q}^{\mathrm{T}}\frac{\partial\mathbf{u}^{\mathrm{c}}}{\partial t}+\mathbf{S}\frac{ \partial\mathbf{p}^{\mathrm{c}}}{\partial t}+\mathbf{P}\mathbf{p}^{\mathrm{c}}=\mathbf{f}_{ \mathrm{p}} \tag{29}\]  
where the storage matrix \(\mathbf{S}\), the permeability matrix \(\mathbf{P}\) and the vector of fluid body forces and fluxes \(\mathbf{f}_{\mathrm{p}}\) are given by  
\[\begin{split}\mathbf{S}&=\int_{\Omega}\mathbf{N}_{\mathrm{p}}^ {\mathrm{T}}c\mathbf{N}_{\mathrm{p}}\mathrm{d}\Omega\\ \mathbf{P}&=\int_{\Omega}\
abla\mathbf{N}_{\mathrm{p}}^{\mathrm{ T}}\frac{1}{\gamma_{t}}\mathbf{k}\
abla\mathbf{N}_{\mathrm{p}}\mathrm{d}\Omega\\ f_{\mathrm{p}}&=\int_{\Omega}\
abla\mathbf{N}_{\mathrm{p }}^{\mathrm{T}}\frac{1}{\gamma_{t}}\mathbf{k}\rho_{\mathrm{f}}\mathrm{d}\Omega-\int _{\Gamma_{N}^{\mathrm{a}}}\mathbf{N}_{Combining Equations (27) and (29) results in the coupled system of equations for poroelasticity  
\[ \begin{bmatrix}\mathbf{0}&\mathbf{0}\\ \mathbf{Q}^{\intercal}&\mathbf{S}\end{bmatrix}\begin{vmatrix}\dot{\mathbf{u}}^{c}\\ \dot{\mathbf{p}}^{c}\end{vmatrix} + \begin{bmatrix}\mathbf{K}&-\mathbf{Q}\\ \mathbf{0}&\mathbf{P}\end{bmatrix}\begin{vmatrix}\mathbf{u}^{c}\\ \mathbf{p}^{c}\end{vmatrix} = \begin{vmatrix}f_{u}\\ f_{p}\end{vmatrix} . \tag{31}\]  
A symmetric system of equations can be obtained by time-differentiating the first equation and multiplying one of the equations by \(-1\), [43]:  
\[ \begin{bmatrix}-\mathbf{K}&\mathbf{Q}\\ \mathbf{Q}^{\intercal}&\mathbf{S}\end{bmatrix}\begin{vmatrix}\dot{\mathbf{u}}^{c}\\ \dot{\mathbf{p}}^{c}\end{vmatrix} + \begin{bmatrix}\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{P}\end{bmatrix}\begin{vmatrix}\mathbf{u}^{c}\\ \mathbf{p}^{c}\end{vmatrix} = \begin{vmatrix}-\dot{\mathbf{f}}_{u}\\ f_{p}\end{vmatrix}  \tag{32}\]  
In this formulation, it is important that time-dependent quantities involved in \(f_{u}\), such as traction and body forces, are \"ramped up\" from an initial equilibrium instead of being applied immediately.