SUMMARYNonphysical pressure oscillations are observed in finite element calculations of Biot's poroelastic equations in low-permeable media. These pressure oscillations may be understood as a failure of compatibility between the finite element spaces, rather than elastic locking. We present evidence to support this view by comparing and contrasting the pressure oscillations in low-permeable porous media with those in lowcompressible porous media. As a consequence, it is possible to use established families of stable mixed elements as candidates for choosing finite element spaces for Biot's equations.

Mathematical models of cardiac electro-mechanics typically consist of three tightly coupled parts: systems of ordinary differential equations describing electro-chemical reactions and cross-bridge dynamics in the muscle cells, a system of partial differential equations modelling the propagation of the electrical activation through the tissue and a nonlinear elasticity problem describing the mechanical deformations of the heart muscle. The complexity of the mathematical model motivates numerical methods based on operator splitting, but simple explicit splitting schemes have been shown to give severe stability problems for realistic models of cardiac electro-mechanical coupling. The stability may be improved by adopting semi-implicit schemes, but these give rise to challenges in updating and linearising the active tension. In this paper we present an operator splitting framework for strongly coupled electro-mechanical simulations and discuss alternative strategies for updating and linearising the active stress component. Numerical experiments demonstrate considerable performance increases from an update method based on a generalised Rush–Larsen scheme and a consistent linearisation of active stress based on the first elasticity tensor.

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Groundwater flow through bounded rectangular aquifers is analyzed in a stochastic framework. New analytical expressions of the head covariance (and variance) and the log transmissivity head cross covariance are derived for the simplified first-order problem. Effects of different boundary conditions, the aquifer size, and the log transmissivity correlation scale upon head moments are studied. The results are compared to analytical half-plane solutions, demonstrating significant differences in several cases. Finally, the problem is solved numerically using the Monte Carlo simulation method. The realizations of the log transmissivity field are generated in two different ways. Through comparisons with the analytical results the accuracy of the numerical methods is shown to be excellent. One of the first contributionsto the field of stochastic groundwater flow was given by Freeze [1975], who solved a one-dimensional problem numerically using Monte Carlo simulation (MCS). A corresponding two-dimensional analysis was performed by Smith and Freeze [1979]. Simplified numerical simulations were made by Sagar [1978] and Dettoenger and Woelson [1981], who adopted a first-order approximation to the flow equation. Analytical first-order solutions were obtained by, for example, Gutjahr and Gelhat [1981], Mizell et al. [1982], Dagan [1979, 1982, 1984], and Towriley and Wilson [1985]. These methods require infinite domains. Lately, Rubin and Dagan [1988, 1989] have written two papers concerning analytical solutions on a half-plane domain with Dirichlet (prescribed head) (1988) and Neumann (no-flow) (1989) conditions imposed on the boundary. Effects of boundaries upon head moments are discussed. The studies above are examples on the so-called direct problem, where moments of the head are obtained for a given variability of the conductivity and initial and boundary conditions. On the other hand, the inverse problem consists in determining moments of the conductivity conditioned on a few measurements of the head and conductivity. Neuman [1980], Carrerra and Neuman [1986], and Loaiciga and Mari•o [1987] have developed numerical solution methods to the problem. Analytical first-order solutions are provided by Dagan [1985a, b] and Rubin and Dagan [1987].While numerical simulations pertain to realistically bounded aquifers, most analytical solutions require infinite or semiinfinite domains. Hence it is difficult to perform thorough comparisons of analytical and numerical results. Therefore one aim of the present study is to develop analytical solutions for bounded aquifers. First-order direct problems will be solved on square and rectangular domains. The analysis offers an opportunity to validate numerical solution methods, and in the last part of the paper results from different MCS methods will be compared to the analytical solutions. To this author's knowledge a similar comparison has not been performed previously.

Uncertainty and variability in material parameters are fundamental challenges in computational biomechanics. Analyzing and quantifying the resulting uncertainty in computed results with parameter sweeps or Monte Carlo methods has become very computationally demanding. In this paper, we consider a stochastic method named the probabilistic collocation method, and investigate its applicability for uncertainty analysis in computing the passive mechanical behavior of the left ventricle. Specifically, we study the effect of uncertainties in material input parameters upon response properties such as the increase in cavity volume, the elongation of the ventricle, the increase in inner radius, the decrease in wall thickness, and the rotation at apex. The numerical simulations conducted herein indicate that the method is well suited for the problem of consideration, and is far more efficient than the Monte Carlo simulation method for obtaining a detailed uncertainty quantification. The numerical experiments also give interesting indications on which material parameters are most critical for accurately determining various global responses.

SUMMARYLarge-scale simulations of flow in deformable porous media require efficient iterative methods for solving the involved systems of linear algebraic equations. Construction of efficient iterative methods is particularly challenging in problems with large jumps in material properties, which is often the case in geological applications, such as basin evolution at regional scales. The success of iterative methods for this type of problems depends strongly on finding effective preconditioners. This paper investigates how the block-structured matrix system arising from single-phase flow in elastic porous media should be preconditioned, in particular for highly discontinuous permeability and significant jumps in elastic properties. The most promising preconditioner combines algebraic multigrid with a Schur complement-based exact block decomposition. The paper compares numerous block preconditioners with the aim of providing guidelines on how to formulate efficient preconditioners.

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Large-scale simulations of coupled flow in deformable porous media require iterative methods for solving the systems of linear algebraic equations. Construction of efficient iterative methods is particularly challenging in problems with large jumps in material properties, which is often the case in realistic geological applications, such as basin evolution at regional scales. The success of iterative methods for such problems depends strongly on finding effective preconditioners with good parallel scaling properties, which is the topic of the present paper. We present a parallel preconditioner for Biot's equations of coupled elasticity and fluid flow in porous media. The preconditioner is based on an approximation of the exact inverse of the two-bytwo block system arising from a finite element discretisation. The approximation relies on a highly scalable approximation of the global Schur complement of the coefficient matrix, combined with generally available state-of-the-art multilevel preconditioners for the individual blocks. This preconditioner is shown to be robust on problems with highly heterogeneous material parameters. We investigate the weak and strong parallel scaling of this preconditioner on up to 512 processors and demonstrate its ability on a realistic basin-scale problem in poroelasticity with over eight million tetrahedral elements.