A new numerically stable sequential algorithm for coupled finite-strain elastoplastic geomechanics and flow

A numerical approach to solve the inverse heat conduction problem

The accurate computation of solutions for multiphase flow problems in a geologically complex subsurface is a great scientific and environmental challenge. Some authors have developed finite-volume methods of new generation (the so-called Multi-Point Flux Approximation methods, commonly called MPFA methods)for addressing this kind of problems.This communication presents some finite-volume based flexible MPFA methods which display strong capabilities to handle flow problems in geologically complex reservoirs.It is well known that the flux continuity across grid-block interfaces combined with the local mass conservation (i.e., mass conservation at grid-block scale)make the finite volume methods a powerful computational tool for flow problems.These properties are met by the variants of MPFA methods presented in this paper. In addition, these new variants provide some added values as (1)an approximate pressure which is continuous in the whole domain, (2)sharp results are provided even on relatively coarse grids, (3)large scale flow problems governed by a symmetric and nonsymmetric full effective permeability tensor can be addressed.A Numerical implementation of these variants of MPFA methods has been done for solving a real-life problem, namely the Darcy flow in the Andra Couplex 1 test case.

A multi-material ALE model for investigating impact dynamics of pile-soil systems

Geometrical nonlinearity arises from the continuous change of shape under increasing loads on thin or long span members, and the materials nonlinearity due to the nonlinear behavior of the materials composing the structural element. In this study the finite element method is used to analyze fibrous reinforced concrete slabs, material and geometric nonlinearities were considered. Concrete is represented by degenerate quadratic thick shell element. Concrete behavior in compression is modeled by elastic perfectly plastic or strain hardening plasticity. A tensile strength criterion is used to follow crack initiation and propagation. A parabolic stress degradation function, which is based on the fracture energy concept, is used to model the tension stiffening effect of steel fibers. The reinforcement is considered as smeared within the concrete layers, and assumed as either an elastic perfectly plastic material or as an elastic-plastic material with linear strain hardening. Von Karman assumptions which is based on the total Lagrangian approach is used for the geometric nonlinearity. The numerical results using for both normal and lightweight fiber reinforced concrete slabs were predicted satisfactorily. The average predicted failure load of the investigated cases to the experimental failure load was 0.899 with a standard deviation of 0.092.

The grid block permeabilities used in reservoir simulation are commonly determined via the upscaling of a fine scale geostatistical reservoir description. Though it is well established that permeabilities computed in this manner are in general full tensor quantities, most finite difference reservoir simulators still treat permeability as a diagonal tensor. In this paper, we implement a capability to handle full tensor permeabilities in a general purpose finite difference simulator and apply this capability to the modeling of several complex geological systems. We formulate a flux continuous approach for the pressure equation using a method analogous to that of previous researchers (Edwards and Rogers; Aavatsmark et al.), consider methods for upwinding in multiphase flow problems, and additionally discuss some relevant implementation and reservoir characterization issues. The accuracy of the finite difference formulation, assessed through comparisons to an accurate finite element approach, is shown to be generally good, particularly for immiscible displacements in heterogeneous systems. The formulation is then applied to the simulation of upscaled descriptions of several geologically complex reservoirs involving crossbedding and extensive fracturing. The method performs quite well for these systems and is shown to accurately capture the effects of the underlying geology. Finally, the significant errors which can be incurred through inaccurate representation of the full permeability tensor are demonstrated for several cases. Introduction Recent advances in reservoir characterization permit the construction of realistic, highly detailed, heterogeneous reservoir descriptions. Such models typically contain far too many grid blocks to simulate directly and therefore require some type of upscaling before they can be used for reservoir simulation. The most important of the upscaled rock properties, for purposes of flow simulation, is the absolute permeability. Accurate procedures for the scale up of permeability generate full tensor permeabilities on the coarse scale, even in cases where the underlying fine scale permeability description is isotropic. Therefore, simulation models generated through scale up of complex reservoir descriptions will in general be characterized by full tensor permeabilities. For many models, however, the off-diagonal components of the effective (or equivalent grid block) permeability tensors can be expected to be small relative to the diagonal components and can generally be ignored. This will typically be the case, for example, when the fine scale permeability is correlated along the coordinate directions (e.g., strictly layered systems) or is correlated nearly isotropically. However, for other types of systems the cross terms of the permeability tensor can be expected to be quite significant. These include formations containing complex crossbedding, dipping layers not aligned with the coordinate system, or extensive fracturing. In these cases the upscaled simulation model will contain full tensor permeabilities with significant off-diagonal terms, which must be accommodated by the simulator if reservoir performance is to be predicted accurately. Our intent here is to develop an approach for the modeling of complex geological systems using a general purpose reservoir simulator. Toward that end, we first develop and implement a formulation for full tensor permeability models, applicable for curvilinear grids, into a general purpose finite difference reservoir simulator. Second, we apply this formulation to the simulation of flow through a variety of upscaled geologic descriptions. P. 253^

Estimation of temperature in rubber-like materials using non-linear finite element analysis based on strain history

We derive two optimal a posteriori error estimators for an implicit fully discrete approximation to the solutions of linear integro-differential equations of the parabolic type. A continuous, piecewise linear finite element space is used for the space discretization and the time discretization is based on an implicit backward Euler method. The a posteriori error indicator corresponding to space discretization is derived using the anisotropic interpolation estimates in conjunction with a Zienkiewicz-Zhu error estimator to approach the error gradient. The error due to time discretization is derived using continuous, piecewise linear polynomial in time. We use the linear approximation of the Volterra integral term to estimate the quadrature error in the second estimator. Numerical experiments are performed on the isotropic mesh to validate the derived results.© 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1309–1330, 2016

A unified high-order Eulerian method for continuum simulations of fluid flow and of elastic–plastic deformations in solids

Applied methods of mathematical modeling, on the basis of which adequate mathematical models and effective calculation technologies that investigate peculiarities of nonlinear deformation of nonhomogeneous thin-walled constructions at different types of static and dynamic loading taking into account geometrical and physical nonlinearity are developed, are represented. One-layer and multilayer shell structures made of both traditional and perspective composite materials are considered. When describing deformation processes for thin-walled structures, mathematical models developed on the basis of Kirchhoff-Love and Timoshenko-Reissner hypothesis are used. Geometrical nonlinearity is considered within the framework of mean-bending theory relations and for taking into account physical nonlinearity relations of deformation plasticity theory with Mises yield criterion supposing elastic unloading are used. Thin-walled shells weakened by cutouts of different forms are considered as well. On external and internal boundaries (cut-out contours) of the shell, different variants of boundary conditions are realized. As loads acting on the shell, a system of boundary and surface loads of general and local character is considered. When solving an original integral-differential problem numerically, the method of finite differences is used for discretization in space and time coordinates. Applying variational-difference method for constructing mesh analogs of equilibrium and motion equations results in conservative difference schemes even for multiconnected regions with complex form of the boundaries. For numerical solutions of stationary problems, the quasi-dynamic form of establishment method is used, which develops calculation algorithms of the same type for solving both static and dynamic problems of nonlinear theory of shells. Relations for determination of the optimal values of iteration process parameters are represented, as is the original technology of critical value of time step significantly decreasing computing time when solving nonlinear problems of shells theory on the basis of explicit difference schemes numerically.

Based on the well-known nonlinear hyperelasticity theory and by using the Carrera Unified Formulation (CUF) as well as a total Lagrangian approach, the unified theory of slightly compressible elastomeric structures including geometrical and physical nonlinearities is developed in this work. By exploiting CUF, the principle of virtual work and a finite element approximation, nonlinear governing equations corresponding to the slightly compressible elastomeric structures are straightforwardly formulated in terms of the fundamental nuclei, which are independent of the theory approximation order. Accordingly, the explicit forms of the secant and tangent stiffness matrices of the unified 1D beam and 2D plate elements are derived by using the three-dimensional Cauchy-Green deformation tensor and the nonlinear constitutive equation for slightly incompressible hyperelastic materials. Several numerical assessments are conducted, including uniaxial tension nonlinear response of rectangular elastomeric beams. Our numerical findings demonstrate the capabilities of the CUF model to calculate the large-deformation equilibrium curves as well as the stress distributions with high accuracy.

New investigations on the method of characteristics for the evaluation of line transients

Singular source terms in sub-diffusion equations may lead to the unboundedness of solutions, which will bring a severe reduction of convergence order of existing time-stepping schemes. In this work, we propose two efficient time-stepping schemes for solving sub-diffusion equations with a class of source terms mildly singular in time. One discretization is based on the Gr{\"u}nwald-Letnikov and backward Euler methods. First-order error estimate with respect to time is rigorously established for singular source terms and nonsmooth initial data. The other scheme derived from the second-order backward differentiation formula (BDF) is proved to possess second-order accuracy in time. Further, piecewise linear finite element and lumped mass finite element discretizations in space are applied and analyzed rigorously. Numerical investigations confirm our theoretical results.

Geometric integration refers to numerical methods which aim to preserve the qualitative and geometric features of a differential equation after discretization. In micromagnetics, the magnetization vector M represents a statistical average of magnetic moments, the magnitude of which should be conserved in time. In general, the midpoint scheme preserves the modulus of solutions on the sphere due to intrinsic quadratic invariance. This method has previously been implemented in various forms to solve the Landau–Lifshitz and Landau–Lifshitz–Gilbert equations within finite-difference formulations. In this article, it is shown that an explicit Euler method will overestimate |M| while an implicit Euler method will make an underestimate, whereas the midpoint method conserves |M| up to round-off error. The midpoint scheme is then utilized within a variational finite-element formulation of the Gilbert equation. Numerical stability and error control are discussed using the reversal of a cobalt nanoparticle as an example calculation.

Constitutive relation error estimators for (visco)plastic finite element analysis with softening

On the discretization in time for a parabolic integrodifferential equation with a weakly singular kernel I: smooth initial data