Harald Osnes

Stochastic analysis of head spatial variability in bounded rectangular heterogeneous aquifers

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.

Computational analysis of geometric nonlinear effects in adhesively bonded single lap composite joints

Abstract It is well known that geometric nonlinear effects have to be taken into account when the ultimate strength of single lap composite joints are studied. In the present paper we investigate for which level of loads or prescribed end displacements nonlinear effects become significant and how they appear. These aspects are studied by comparing finite element results obtained from geometric nonlinear models with the results from the linear ones. The well-known software package ANSYS is applied in the numerical analysis together with a self-implemented module in the C++ library Diffpack. Some of the results are also compared with classical analytical theories of idealized joints showing significant differences. The joints examined are made of cross-ply laminates having 0 or 90° surface layers. A combined cross-ply/steel joint and an isotropic joint made of steel are also studied. All the models except the all-steel one are assembled with adhesives, while the latter is welded. Through the investigation a considerable departure from linear behavior has been detected for a large regime of prescribed end displacements or external loads. Geometric nonlinear effects begin to develop for external loads that produces stresses which are far below ultimate strength limits and for average longitudinal strains that are less than 0.5%. It has also been detected that the distribution of materials within the joint has some influence on the nonlinear behavior. Thus, geometric nonlinear methods should always be applied when single lap (or other non-symmetric) composite joints are analyzed.

Stochastic analysis of velocity spatial variability in bounded rectangular heterogeneous aquifers

Transport of inert solutes in two-dimensional bounded heterogeneous porous media is investigated in a stochastic framework. After adopting a first-order approximation of the flow equations, analytical expressions are derived for the velocity covariances. Effects of the boundary conditions and aquifer size upon the statistical moments are analyzed. While the size of the domain is shown to have small influence on the covariances in most cases, the solutions are considerably modified by the boundaries. The results are compared with analytical solutions on infinite domains, and several discrepancies are demonstrated. For example, while the velocity variances on infinite domains are homogeneous, the present results are strongly non-stationary. Finally, the problem is solved numerically by the Monte Carlo simulation method. The results, including the behavior near the boundaries, are shown to be in close agreement with analytical solutions.

An efficient probabilistic finite element method for stochastic groundwater flow

We present an efficient numerical method for solving stochastic porous media flow problems. Single-phase flow with a random conductivity field is considered in a standard first-order perturbation expansion framework. The numerical scheme, based on finite element techniques, is computationally more efficient than traditional approaches because one can work with a much coarser finite element mesh. This is achieved by avoiding the common finite element representation of the conductivity field. Computations with the random conductivity field only arise in integrals of the log conductivity covariance function. The method is demonstrated in several two- and three-dimensional flow situations and compared to analytical solutions and Monte Carlo simulations. Provided that the integrals involving the covariance of the log conductivity are computed by higher-order Gaussian quadrature rules, excellent results can be obtained with characteristic element sizes equal to about five correlation lengths of the log conductivity field. Investigations of the validity of the proposed first-order method are performed by comparing nonlinear Monte Carlo results with linear solutions. In box-shaped domains the log conductivity standard deviation σY may be as large as 1.5, while the head variance is considerably influenced by nonlinear effects as σY approaches unity in more general domains.

Improved discretisation and linearisation of active tension in strongly coupled cardiac electro-mechanics simulations

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.

Uncertainty Analysis of Ventricular Mechanics Using the Probabilistic Collocation Method

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.

A Study of Some Finite Difference Schemes for a Unidirectional Stochastic Transport Equation

We study a unidirectional hyperbolic transport equation, with a homogeneous stochastic transport velocity, solved by Monte Carlo simulation. Several finite difference schemes are applied to the deterministic problem in each Monte Carlo iteration, and the numerical solution of the stochastic problem is compared to analytical solutions derived in the paper. We present both a theoretical analysis and summarized results from extensive numerical experiments. The behavior of the various schemes depends on the stochastic properties of the problem, and there are new demands on the schemes when they are used as part of a Monte Carlo simulation. For example, schemes that are very oscillatory for a single deterministic problem, like the leapfrog scheme, turn out to be efficient and accurate for the corresponding stochastic problem.

A parallel block preconditioner for large-scale poroelasticity with highly heterogeneous material parameters

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-by-two 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.

A mixed finite element formulation for a non-linear, transversely isotropic material model for the cardiac tissue

In this paper we present a mixed finite element method for modeling the passive properties of the myocardium. The passive properties are described by a non-linear, transversely isotropic, hyperelastic material model, and the myocardium is assumed to be almost incompressible. Single-field, pure displacement-based formulations are known to cause numerical difficulties when applied to incompressible or slightly compressible material cases. This paper presents an alternative approach in the form of a mixed formulation, where a separately interpolated pressure field is introduced as a primary unknown in addition to the displacement field. Moreover, a constraint term is included in the formulation to enforce (almost) incompressibility. Numerical results presented in the paper demonstrate the difficulties related to employing a pure displacement-based method, applying a set of physically relevant material parameter values for the cardiac tissue. The same problems are not experienced for the proposed mixed method. We show that the mixed formulation provides reasonable numerical results for compressible as well as nearly incompressible cases, also in situations of large fiber stretches. There is good agreement between the numerical results and the underlying analytical models.

Strength of bonded overlap composite joints in marine applications

Abstract: This chapter deals with methods for predicting the strength of overlap composite joints for marine applications such as ships and offshore structures. Failure loads obtained experimentally are presented and compared with theoretical predictions. Capacity estimates provided by traditional strength of materials approaches do not agree with the experiments, but results obtained using a recently developed inelastic fracture-based analysis method represent the measured strength values well.