# Mixed Method for Isogeometric Analysis of Coupled Flow and Deformation in Poroelastic Media
## 5 Conclusions  
Mixed isogeometric analysis of poroelasticity is presented where different orders of polynomials are used for the displacement and pore pressure field variables. Numerical studies on [PERSON]'s classical one-dimensional consolidation problem and consolidation of a layered soil with a middle low permeability layer are presented. The results from mixed polynomial order simulations are compared with equal order analyses.  
For [PERSON]'s one-dimensional consolidation problem, the pore pressure oscillations are investigated when a time step size less than the critical value is used. The oscillations were observed to be higher in the equal order simulations compared to the mixed order results. The oscillations are not completely removed in the mixed isogeometric simulations but it is observed that they tend to decrease with increasing polynomial orders for the pore pressure. This is illustrated by the convergence of the relative \(L^{2}\) norm of the pore pressure error for varying polynomial orders.  
The low permeability layer problem showed similar trends in the pore pressure oscillations, i.e., the equal order simulations resulted in worse pore pressure oscillations compared to the mixed results. Again, in both cases, the oscillations decreased with increasing polynomial orders. The use of a graded mesh, where the knot spans are concentrated at the interfaces between the low permeability and other layers, resulted in much lower oscillations both in the equal order and mixed cases. This indicates the potential of adaptive refinement for such class of problems.  
Our isogeometric mixed finite elements with maximum continuity show superior results compared to earlier studies of these problems using C\({}^{0}\)-continuous mixed Lagrange finite elements. In particular, for the problem of consolidation of a very low permeability layer, the improved performance of the isogeometric finite element formulation (due to the higher regularity) is significant.  
Conceptualization, Y.W.B., E.F. and T.K.; methodology, Y.W.B., E.F. and T.K.; software, Y.W.B., E.F. and A.M.K.; validation, Y.W.B. and E.F.; formal analysis, Y.W.B. and E.F.; investigation, Y.W.B. and E.F.; resources, Y.W.B., E.F., T.K., A.M.K. and S.N.; data curation, Y.W.B., E.F. and T.K.; writing--original draft preparation, Y.W.B. and E.F.; writing--review and editing, Y.W.B., E.F. and T.K.; visualization, Y.W.B. and E.F.; supervision, T.K. and S.N.; project administration, T.K. and S.N.; funding acquisition, S.N. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Research Council of Norway and industrial partners through the research project SAMCoT, Sustainable Arctic Marine and Coastal Technology, project number 203471.  
Not applicable.  
The numerical work in this study is implemented using IFEM, an open-source isogeometric analysis software developed at SINTEF, as a basis. The authors would like to acknowledge the IFEM software team at SINTEF and other contributors.  
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.  
The following abbreviations are used in this manuscript:  
CAD Computer-Aided Design  
FEA Finite Element Analysis  
GTR Generalized Trapezoidal Rule  
IGA Isogeometric Analysis  
NURBS Non-uniform Rational Basis Spline