Finite element solutions for nonhomogeneous nonlocal elastic problems

A finite element procedure for analysing nonhomogeneous nonlocal elastic 2D problems is presented and discussed. The procedure grounds on a variationally consistent approach known, in the relevant literature, as Nonlocal Finite Element Method. The latter is recast making use of a recently theorized phenomenological strain-difference-based nonhomogeneous nonlocal elastic model. The peculiarities of the numerical procedure together with the pertinent nonlocal operators are expounded and discussed. Two simple numerical 2D examples close the paper.     

SOLUTION OF THE ADVECTION–DIFFUSION EQUATION USING A COMBINATION OF DISCONTINUOUS AND MIXED FINITE ELEMENTS

When transport is advection-dominated, classical numerical methods introduce excessive artificial diffusion and spurious oscillations. Special methods are required to overcome these phenomena. To solve the advection‒diffusion equation, a numerical method is developed using a discontinuous finite element method for the discretization of the advective terms. At the discontinuities of the approximate solution, numerical advective fluxes are calculated using one-dimensional approximate Riemann solvers. The method is stabilized with a multidimensional slope limiter which introduces small amounts of numerical diffusion when sharp concentration fronts occur. In addition, the diffusive term is discretized using a mixed hybrid finite element method. With this approach, numerical oscillations are completely avoided for a full range of cell Peclet numbers. The combination of discontinuous and mixed finite elements can be easily applied to 2D and 3D models using various types of elements in regular and irregular meshes. Numerical tests show good agreement with 1D and 2D analytical solutions. This approach is compared at the same time with two different numerical methods, a standard mixed finite method and a finite volume approach with high-resolution upwind terms. Regular and irregular meshes are used for the numerical tests to study the mesh effects on the numerical results. Our data show that in all cases this approach performs well. © 1997 by John Wiley & Sons, Ltd.     

Advective transport in a multilayered system of aquifers

A three-dimensional model of transport in porous media, consisting of several aquifers and aquitards, is presented. In the solution procedure a two-dimensional flow model has been adapted to incorporate three-dimensional velocity components. This procedure enables to observe the way particles of a pollutant are being transported through the system of aquifers and aquitards, while maintaining the Dupuit assumption of zero vertical gradients of the hydraulic head in the aquifers. The governing equations are solved using a finite element technique. The transport of pollutants is restricted to advective transport and linear adsorption.     

Introducing new cracked finite elements and a method for SIF calculation of cracks

Two different types of 8-node cracked quadrilateral finite element are presented for fracture applications. The first element contains a central crack and the other one includes an edge crack. The introduced elements are applicable in 2D problems. The crack is not physically modeled within the element, but instead, its effects on the stiffness matrix are taken into account by utilizing linear fracture mechanics laws. Furthermore, a simple and practical procedure is proposed for calculation of stress intensity factor (SIF) by employing proposed cracked elements. Several numerical examples are presented to evaluate the capabilities of the proposed elements and procedure.     

Modeling of cementitious materials exposed to isothermal calcium leaching,considering process kinetics and advective water flow. Part 2: Numerical solution

The second part of the paper presents numerical solutions of the mathematical model of hydro-chemo-mechanical behavior of cementitious materials exposed to contact with deionized water of part 1. The model defines kinetics of the calcium leaching process instead of a direct application of a curve describing equilibrium between solid calcium in the material skeleton and the calcium dissolved in the pore solution. It further takes into account the advective flux of calcium ions. Both aspects are new as compared to previous models. The weak form of the governing equations of the model is derived first using the Galerkin method. Then, the equations are discretized in space with finite elements and in time domain with finite differences, and finally the procedures used for numerical solution of their discretized form are presented. Three numerical examples are solved to test the numerical solution procedure proposed and demonstrate its robustness for solution of 1D and 2D problems concerning fast and slow leaching of cement-based materials. The effect of various factors on the results concerning chemical degradation of structures made of cementitious materials is analyzed as well.     

A finite element formulation satisfying the discrete geometric conservation law based on averaged Jacobians

In this article, a new methodology for developing discrete geometric conservation law (DGCL) compliant formulations is presented. It is carried out in the context of the finite element method for general advective–diffusive systems on moving domains using an ALE scheme. There is an extensive literature about the impact of DGCL compliance on the stability and precision of time integration methods. In those articles, it has been proved that satisfying the DGCL is a necessary and sufficient condition for any ALE scheme to maintain on moving grids the nonlinear stability properties of its fixed‐grid counterpart. However, only a few works proposed a methodology for obtaining a compliant scheme. In this work, a DGCL compliant scheme based on an averaged ALE Jacobians formulation is obtained. This new formulation is applied to the θ family of time integration methods. In addition, an extension to the three‐point backward difference formula is given. With the aim to validate the averaged ALE Jacobians formulation, a set of numerical tests are performed. These tests include 2D and 3D diffusion problems with different mesh movements and the 2D compressible Navier–Stokes equations. Copyright © 2011 John Wiley & Sons, Ltd.     

Poromechanical Modelling and Inverse Approach of Drying Tests on Weakly Permeable Porous Rocks

The present paper deals with the determination of permeability in partially saturated conditions for weakly permeable porous rocks such as argillites or deep clayey formations. The level of permeability can be obtained via the measurements of transient weight loss of a sample submitted to a decrease in relative humidity imposed by saline solution in a hermetic chamber. An identification method based on simplified uncoupled 1D-linear and 1D-non-linear modelling was presented in a previous paper (Giraud et al. Trans Porous Media 69(2):259–280, 2006). The present paper takes into account generalized mass transfer phenomena such as Darcean advective transport of liquid and gas mixtures and Fickean diffusive transport of the vapour specie inside a gas mixture. Poromechanical coupling as well as 3D effects due to the geometry and finite dimensions of the tested samples are also covered by this approach. The coupled THM finite element computer code Code_Aster is then used to model the forward problem. The parameter identification procedure is based upon the solution of an inverse problem. The Levenberg–Marquardt algorithm was used for the problem of minimization. Comparisons between previous simplified 1D modelling and 2D-axisymmetrical coupled modelling show that the former method efficiently provides the correct order of magnitude of the level of permeability or the equivalent storage coefficients. Due to the boundary condition, the real 2D-axisymmetrical geometry of the sample must not be neglected if we are to obtain accurate results.     

Lagrangian finite element analysis applied to viscous free surface fluid flow

A new Lagrangian finite element formulation is presented for time-dependent incompressible free surface fluid flow problems described by the Navier-Stokes equations. The partial differential equations describing the continuum motion of the fluid are discretized using a Galerkin procedure in conjunction with the finite element approximation. Triangular finite elements are used to represent the dependent variables of the problem. An effective time integration procedure is introduced and provides a viable computational method for solving problems with equality of representation of the pressure and velocity fields. Its success has been attributed to the strict enforcement of the continuity constraint at every stage of the iterative process. The capabilities of the analysis procedure and the computer programs are demonstrated through the solution of several problems in viscous free surface fluid flow. Comparisons of results are presented with previous theoretical, numerical and experimental results.     

Recursive super-convergence computation for multi-dimensional problems via one-dimensional element energy projection technique

This paper presents a strategy for computation of super-convergent solutions of multi-dimensional problems in the finite element method (FEM) by recursive application of the one-dimensional (1D) element energy projection (EEP) technique. The main idea is to conceptually treat multi-dimensional problems as generalized 1D problems, based on which the concepts of generalized 1D FEM and its consequent EEP formulae have been developed in a unified manner. Equipped with these concepts, multi-dimensional problems can be recursively discretized in one dimension at each step, until a fully discretized standard finite element (FE) model is reached. This conceptual dimension-by-dimension (D-by-D) discretization procedure is entirely equivalent to a full FE discretization. As a reverse D-by-D recovery procedure, by using the unified EEP formulae together with proper extraction of the generalized nodal solutions, super-convergent displacements and first derivatives for two-dimensional (2D) and three-dimensional (3D) problems can be obtained over the domain. Numerical examples of 3D Poisson’s equation and elasticity problem are given to verify the feasibility and effectiveness of the proposed strategy.     

Simulations of 2D and 3D thermocapillary flows by a least-squares finite element method

Numerical results for time-dependent 2D and 3D thermocapillary flows are presented in this work. The numerical algorithm is based on the Crank–Nicolson scheme for time integration, Newton's method for linearization, and a least-squares finite element method, together with a matrix-free Jacobi conjugate gradient technique. The main objective in this work is to demonstrate how the least-squares finite element method, together with an iterative procedure, deals with the capillary-traction boundary conditions at the free surface, which involves the coupling of velocity and temperature gradients. Mesh refinement studies were also carried out to validate the numerical results. © 1998 John Wiley & Sons, Ltd.     

Two-equation (k, ϵ) turbulence computations by the use of a finite element model

A finite element technique is presented and applied to some one- and two-dimensional turbulent flow problems. The basic equations are the Reynolds averaged momentum equations in conjunction with a two-equation (k, ?) turbulence model. The equations are written in time-dependent form and stationary problems are solved by a time iteration procedure. The advection parts of the equations are treated by the use of a method of characteristics, while the continuity requirement is satisfied by a penalty function approach. The general numerical formulation is based on Galerkin's method. Computational results are presented for one-dimensional steady-state and oscillatory channel flow problems and for steady-state flow over a two-dimensional backward-facing step.     

Finite element simulation of penetration into geological targets

The finite element method has been applied to study the projectile penetration process into geological targets. To illustrate the solution procedure, two example problems involving conical nose steel penetrators into sea ice and antelope tuff are presented. Details of the numerical treatment using the computer code PRONTO 2D are given. Good agreements between numerical and experimental results are observed. Although the examples are limited to normal impact problems in which the axis of the penetrator is perpendicular to the target surface and where no angle of attack for the penetrator is allowed, some thoughts regarding ways to deal with nonnormal three-dimensional penetration problems are discussed.     

Numerical modelling of shear-dependent mass transfer in large arteries

A numerical scheme for the simulation of blood flow and transport processes in large arteries is presented. Blood flow is described by the unsteady 3D incompressible Navier–Stokes equations for Newtonian fluids; solute transport is modelled by the advection–diffusion equation. The resistance of the arterial wall to transmural transport is described by a shear-dependent wall permeability model. The finite element formulation of the Navier–Stokes equations is based on an operator-splitting method and implicit time discretization. The streamline upwind/Petrov–Galerkin (SUPG) method is applied for stabilization of the advective terms in the transport equation and in the flow equations. A numerical simulation is carried out for pulsatile mass transport in a 3D arterial bend to demonstrate the influence of arterial flow patterns on wall permeability characteristics and transmural mass transfer. The main result is a substantial wall flux reduction at the inner side of the curved region. © 1997 John Wiley & Sons, Ltd.     

An h-adaptive solution of the spherical blast wave problem

Shock waves and contact discontinuities usually appear in compressible flows, requiring a fine mesh in order to achieve an acceptable accuracy of the numerical solution. The usage of a mesh adaptation strategy is convenient as uniform refinement of the whole mesh becomes prohibitive in three-dimensional (3D) problems. An unsteady h-adaptive strategy for unstructured finite element meshes is introduced. Non-conformity of the refined mesh and a bounded decrease in the geometrical quality of the elements are some features of the refinement algorithm. A 3D extension of the well-known refinement constraint for 2D meshes is used to enforce a smooth size transition among neighbour elements with different levels of refinement. A density-based gradient indicator is used to track discontinuities. The solution procedure is partially parallelised, i.e. the inviscid flow equations are solved in parallel with a finite element SUPG formulation with shock capturing terms while the adaptation of the mesh is sequentially performed. Results are presented for a spherical blast wave driven by a point-like explosion with an initial pressure jump of 10⁵ atmospheres. The adapted solution is compared to that computed on a fixed mesh. Also, the results provided by the theory of self-similar solutions are considered for the analysis. In this particular problem, adapting the mesh to the solution accounts for approximately 4% of the total simulation time and the refinement algorithm scales almost linearly with the size of the problem.     

Numerical investigation of the slope discontinuities in large deformation finite element formulations

In many multibody system applications, the system components are made of structural elements that can have different orientations, leading to slope discontinuities. In this paper, a numerical investigation of a new procedure that can be used to model structures with slope discontinuities in the finite element absolute nodal coordinate formulation (ANCF) is presented. This procedure can be applied to model slope discontinuities in the case of commutative rotations of gradient deficient elements that are used for modeling thin beam and plate structures. An important special case to which the proposed procedure can be applied is the case of all planar gradient deficient ANCF finite elements. The use of the proposed method leads to a constant orthogonal element transformation that describes an arbitrary initial configuration. As a consequence, one obtains, in the case of large commutative rotations and large deformations, a constant mass matrix for structures which have complex geometry. The procedure used in this investigation to model slope discontinuities requires the use of the concept of the intermediate finite element coordinate system. For each finite element, a new set of gradient coordinates that define, at the discontinuity node, the element deformation with respect to the intermediate element coordinate system is introduced. These new gradient coordinates are assumed to be equal for the two finite elements at the point of intersection. That is, the change of the gradients of two elements at the intersection point from their respective intermediate initial reference configuration is assumed to be the same. This procedure leads to a set of linear algebraic equations that define the orthogonal transformation matrix for the finite element. Numerical examples are presented in order to demonstrate the use of the proposed procedure for modeling slope discontinuities.     

Bending of beams on three-parameter elastic foundation

The bending of a Timoshenko beam resting on a Kerr-type three-parameter elastic foundation is introduced, its governing differential equations are formulated and analytically solved, and the solutions are discussed and applied to particular problems. Parametric analyses of elastically supported beams of infinite and finite length are carried out and comparisons are made between one, two or three-parameter foundation models and more accurate 2D finite element models. In order to estimate the necessary soil parameters, an analytical procedure based on the modified Vlasov model is proposed. The presented solutions and applications show the superiority of the Kerr-type foundation model compared to one or two-parameter models.     

Crack insertion,meshing and fracture analysis of structures using tetrahedral finite elements

In this study, a method and corresponding tools are presented to insert a three-dimensional crack of a given size and location into a finite element model without any cracks using fully unstructured finite elements. For research purposes, publicly available two and three-dimensional meshing software, Triangle^© and Tetgen^©, are utilized and integrated with an in-house developed program to compatibly select and re-mesh the three-dimensional crack region of the original input model. Within the procedure, the boundary conditions and loads existing on the original model are also book kept and transferred to the new model containing the crack. Next, the new finite element model, which now contains the crack geometry, the loads and boundary conditions, is solved in a general-purpose finite element program employing enriched elements. The above procedure is demonstrated on a series of surface crack problems in finite-thickness plates including mixed-mode fracture conditions. The obtained results are compared to well-known solutions available in the literature. These comparisons showed good agreement for all cases analyzed. It is, therefore, concluded that the procedure developed is valid, efficient and yields accurate three-dimensional fracture solutions.     

The effect of numerical methods on the simulation of mid-ocean ridge hydrothermal models

This work considers the effect of the numerical method on the simulation of a 2D model of hydrothermal systems located in the high-permeability axial plane of mid-ocean ridges. The behavior of hot plumes, formed in a porous medium between volcanic lava and the ocean floor, is very irregular due to convective instabilities. Therefore, we discuss and compare two different numerical methods for solving the mathematical model of this system. In concrete, we consider two ways to treat the temperature equation of the model: a semi-Lagrangian formulation of the advective terms in combination with a Galerkin finite element method for the parabolic part of the equations and a stabilized finite element scheme. Both methods are very robust and accurate. However, due to physical instabilities in the system at high Rayleigh number, the effect of the numerical method is significant with regard to the temperature distribution at a certain time instant. The good news is that relevant statistical quantities remain relatively stable and coincide for the two numerical schemes. The agreement is larger in the case of a mathematical model with constant water properties. In the case of a model with nonlinear dependence of the water properties on the temperature and pressure, the agreement in the statistics is clearly less pronounced. Hence, the presented work accentuates the need for a strengthened validation of the compatibility between numerical scheme (accuracy/resolution) and complex (realistic/nonlinear) models.     

The Runge–Kutta control volume discontinuous finite element method for systems of hyperbolic conservation laws

In this paper, a new high‐order and high‐resolution method called the Runge–Kutta control volume discontinuous finite element method (RKCVDFEM) was proposed to solve 1D and 2D systems of hyperbolic conservation laws. Its main advantage lies in the local conservation, and it is simpler than the Runge–Kutta discontinuous Galerkin finite element method (RKDGM). The theoretical analysis showed that the RKCVDFEM has formally an optimal convergence order for 1D systems. Based on logically rectangular grids of irregular quadrilaterals, a scheme for 2D systems was constructed. Some classical problems were simulated and the validity of the method was presented. Copyright © 2010 John Wiley & Sons, Ltd.     

Normal flow boundary conditions in 3D circulation models

The problem of enforcing normal transport conditions on 3D velocity fields is considered in the context of ‘wave equation’ finite element models. A procedure for strong enforcement of the transport constraint is given. The procedure is identical for Neumann (transport known) and Dirichlet (pressure known) problems, which are treated reversibly. All local mass and force balance relations are retained in the FEM system. A global mass conservation property is proven for the general 3D, discrete-time case. Examples demonstrate the quality of the solutions and the practicality of the approach. © 1997 John Wiley & Sons, Ltd.