A Stress/Displacement Virtual Element Method for Plane Elasticity Problems

The numerical approximation of 2D elasticity problems is considered, in the framework of the small strain theory and in connection with the mixed Hellinger-Reissner variational formulation. A low-order Virtual Element Method (VEM) with a-priori symmetric stresses is proposed. Several numerical tests are provided, along with a rigorous stability and convergence analysis

A finite element analysis of crack propagation problems with applications to seismology

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A Comparison of Numerical Methods used for\ud Finite Element Modelling of Soft Tissue\ud Deformation

Soft tissue deformation is often modelled using incompressible nonlinear elasticity, with solutions computed using the finite element method. There are a range of options available when using the finite element method, in particular, the polynomial degree of the basis functions used for interpolating position and pressure, and the type of element making up the mesh. We investigate the effect of these choices on the accuracy of the computed solution, using a selection of model problems motivated by typical deformations seen in soft tissue modelling. We set up model problems with discontinuous material properties (as is the case for the breast), steeply changing gradients in the body force (as found in contracting cardiac tissue), and discontinuous first derivatives in the solution at the boundary, caused by a discontinuous applied force (as in the breast during mammography). We find that the choice of pressure basis functions are vital in the presence of a material interface, higher-order schemes do not perform as well as may be expected when there are sharp gradients, and in general that it is important to take the expected regularity of the solution into account when choosing a numerical scheme

A Stabilized RBF Collocation Scheme for Neumann Type Boundary Value Problems

The numerical solution of partial differential equations (PDEs) with Neumann boundary conditions (BCs) resulted from strong form collocation scheme are typically much poorer in accuracy compared to those with pure Dirichlet BCs. In this paper, we show numerically that the reason of the reduced accuracy is that Neumann BC requires the approximation of the spatial derivatives at Neumann boundaries which are significantly less accurate than approximation of main function. Therefore, we utilize boundary treatment schemes that based upon increasing the accuracy of spatial derivatives at boundaries. Increased accuracy of the spatial derivative approximation can be achieved by h-refmement reducing the spacing between discretization points or by increasing the multiquadric shape parameter, c. Increasing the MQ shape parameter is very computationally cost effective, but leads to increased ill-conditioning. We have implemented an improved version of the truncated singular value decomposition (IT-SVD) originated by Volokh and Vilnay (2000) that projects very small singular values into the null space, producing a well conditioned system of equations. To assess the proposed refinement scheme, elliptic PDEs with different boundary conditions are analyzed. Comparisons that made with analytical solution reveal superior accuracy and computational efficiency of the IT-SVD solutions

Mixed Formulations in Space and Time Discretizations for the Fixed-Stress Split Method in Poromechanics

Coupled flow and geomechanics become one of the important research topics in oil and gas industry for development of unconventional petroleum reservoirs such as gas shale, tight gas, and gas hydrates. In particular, these reservoirs are naturally born with its complex behavior, exhibiting strong non-linearity, anisotropy, and heterogeneity effects within each geomaterial and fluid by itself. In addition, the coupling between flow and geomechanics is more complicated for unconsolidated reservoirs or shale formations. Thus, it is critical to assess these complex coupled processes properly through poromechanics with forward numerical simulation and to provide more accurate solutions in order to predict the reservoir performance more precisely. The main objective of this study is to address several numerical issues that are accompanied with simulation in poromechanics. We perform in-depth analysis on mathematical conditions to satisfy for numerically stable and accurate solution, employing various mixed formulations in space and time discretization. Specifically, in space discretization, we deal with the spatial instability that occurs at early times in poromechanics simulation, such as a consolidation problem. We identify two types of spatial instabilities caused by violation of two different conditions: the condition due to discontinuity in pressure and the inf-sup condition related to incompressible fluid, which both occur at early times. We find that the fixed-stress split with the finite volume method for flow and finite element method for geomechanics can provide stability in space, allowing discontinuity of pressure and circumventing violation of the inf-sup condition. In time discretization, we investigate the order of accuracy in time integration with the fixed-stress sequential method. In the study, two-pass and deferred correction methods are to be considered for studying the high-order methods in time integration. We find that there are two different inherent constraint structures that still cause order reductions against high-order accuracy while applying the two methods. As an additional in-depth analysis, we study a large deformation system, considering anisotropic properties for geomechanical and fluid flow parameters, the traverse isotropy and permeability anisotropy ratio. Seeking more accurate solutions, we adopt the total Lagrangian method in geomechanics and multi-point flux approximation in fluid flow. By comparing it to the infinitesimal transformation with two-point flux approximation, we find that substantial differences between the two approaches can exist. For a field application, we study large-scale geomechanics simulation that can honor measured well data, which leads to a constrained geomechanics problem. We employ the Uzawa’s algorithm to solve the saddle point problem from the constrained poromechanics. From numerical parallel simulations, we estimate initial stress distribution in the shale gas reservoir, which will be used for the field development plan. From this study, we find several mathematical conditions for numerically stable and accurate solution of poromechanics problems, when we take the various mixed formulations. By considering the conditions, we can overcome the numerical issues. Then, reliable and precise prediction of reservoir behavior can be obtained for coupled flow-geomechanics problems

Mixed Formulations in Space and Time Discretizations for the Fixed-Stress Split Method in Poromechanics

Coupled flow and geomechanics become one of the important research topics in oil and gas industry for development of unconventional petroleum reservoirs such as gas shale, tight gas, and gas hydrates. In particular, these reservoirs are naturally born with its complex behavior, exhibiting strong non-linearity, anisotropy, and heterogeneity effects within each geomaterial and fluid by itself. In addition, the coupling between flow and geomechanics is more complicated for unconsolidated reservoirs or shale formations. Thus, it is critical to assess these complex coupled processes properly through poromechanics with forward numerical simulation and to provide more accurate solutions in order to predict the reservoir performance more precisely. The main objective of this study is to address several numerical issues that are accompanied with simulation in poromechanics. We perform in-depth analysis on mathematical conditions to satisfy for numerically stable and accurate solution, employing various mixed formulations in space and time discretization. Specifically, in space discretization, we deal with the spatial instability that occurs at early times in poromechanics simulation, such as a consolidation problem. We identify two types of spatial instabilities caused by violation of two different conditions: the condition due to discontinuity in pressure and the inf-sup condition related to incompressible fluid, which both occur at early times. We find that the fixed-stress split with the finite volume method for flow and finite element method for geomechanics can provide stability in space, allowing discontinuity of pressure and circumventing violation of the inf-sup condition. In time discretization, we investigate the order of accuracy in time integration with the fixed-stress sequential method. In the study, two-pass and deferred correction methods are to be considered for studying the high-order methods in time integration. We find that there are two different inherent constraint structures that still cause order reductions against high-order accuracy while applying the two methods. As an additional in-depth analysis, we study a large deformation system, considering anisotropic properties for geomechanical and fluid flow parameters, the traverse isotropy and permeability anisotropy ratio. Seeking more accurate solutions, we adopt the total Lagrangian method in geomechanics and multi-point flux approximation in fluid flow. By comparing it to the infinitesimal transformation with two-point flux approximation, we find that substantial differences between the two approaches can exist. For a field application, we study large-scale geomechanics simulation that can honor measured well data, which leads to a constrained geomechanics problem. We employ the Uzawa’s algorithm to solve the saddle point problem from the constrained poromechanics. From numerical parallel simulations, we estimate initial stress distribution in the shale gas reservoir, which will be used for the field development plan. From this study, we find several mathematical conditions for numerically stable and accurate solution of poromechanics problems, when we take the various mixed formulations. By considering the conditions, we can overcome the numerical issues. Then, reliable and precise prediction of reservoir behavior can be obtained for coupled flow-geomechanics problems

Computational Engineering

The focus of this Computational Engineering Workshop was on the mathematical foundation of state-of-the-art and emerging finite element methods in engineering analysis. The 52 participants included mathematicians and engineers with shared interest on discontinuous Galerkin or Petrov-Galerkin methods and other generalized nonconforming or mixed finite element methods

A lumped stress method for plane elastic problemsand the discrete-continuum approximation

This paper proposes a rational method to approximate a plane elastic body through a latticed structure composed of truss elements. The method is based on the introduction of a relaxed stress energy that allows an extension of the original problem to a larger space of admissible stress fields, including stresses concentrated along lines. Use is made of polyhedral approximations of the Airy stress function. The truss analogy is employed to obtain a displacement formulation. The paper includes several numerical applications of the method to sample problems, a numerical convergence study and comparisons with exact solutions and standard finite element approximations