A finite element approach for multiphase fluid flow in porous media

The main scope of this work is to carry out a mathematical framework and its corresponding finite element (FE) discretization for the partially saturated soil consolidation modelling in presence of an immiscible pollutant. A multiphase system with the interstitial voids in the grain matrix filled with water (liquid phase), water vapour and dry air (gas phase) and with pollutant substances, is assumed. The mathematical model addressed in this work was developed in the framework of mixture theory considering the pollutant saturation-suction coupling effects. The ensuing mathematical model involves equations of momentum balance, energy balance and mass balance of the whole multiphase system. Encouraging outcomes were achieved in several different examples.     

A finite element alternative to infinite elements

In this paper, a simple idea based on midpoint integration rule is utilized to solve a particular class of mechanics problems; namely static problems defined on unbounded domains where the solution is required to be accurate only in an interior (and not in the far field). By developing a finite element mesh that approximates the stiffness of an unbounded domain directly (without approximating the far-field displacement profile first), the current formulation provides a superior alternative to infinite elements (IEs) that have long been used to incorporate unbounded domains into the finite element method (FEM). In contrast to most IEs, the present formulation (a) requires no new shape functions or special integration rules, (b) is proved to be both accurate and efficient, and (c) is versatile enough to handle a large variety of domains including those with anisotropic, stratified media and convex polygonal corners. In addition to this, the proposed model leads to the derivation of a simple error expression that provides an explicit correlation between the mesh parameters and the accuracy achieved. This error expression can be used to calculate the accuracy of a given mesh a-priori. This in-turn, allows one to generate the most efficient mesh capable of achieving a desired accuracy by solving a mesh optimization problem. We formulate such an optimization problem, solve it and use the results to develop a practical mesh generation methodology. This methodology does not require any additional computation on the part of the user, and can hence be used in practical situations to quickly generate an efficient and near optimal finite element mesh that models an unbounded domain to the required accuracy. Numerical examples involving practical problems are presented at the end to illustrate the effectiveness of this method.     

A finite element approach to global-local modeling in composite laminate analysis

A finite element model is described to study interlaminar stresses within polymer composite laminated materials. This model is based upon a global-local model proposed by Pagano and Soni in 1983. The development of solution procedures includes an out-of-core memory solving technique. The numerical results generated for simple plate problems with and without holes in the center under uniaxial loading are reported. Comments regarding the finite-element mesh-size, numerical stability, problem size and sensitivity of results to substructuring of the laminate into global and local regions have also been discussed.     

A finite element method for flow separation

The finite element model has been developed in order to solve separation pattern of the flow past an obstruction in a two-dimensional flow field. The Helmholtz-Poisson form of the Reynolds equations are solved alternately until a stable flow separation in the neighbourhood of the obstruction is obtained. In order to check the results of the finite element model, an experimental separation pattern using Pitot-tube measurements has been conducted. The computed and the experimental flow separation patterns show a good agreement.     

A transient finite element solution method for the Navier-Stokes equations

This paper is concerned with the discrete finite element formulation and numerical solution of transient incompressible viscous flow in terms of the primitive variables. A restricted variational principle is introduced as equivalent to the momentum equations and the Poisson equation for pressure. The latter is introduced to replace the continuity equation, and thus the incompressibility condition is realized only asymptotically; i.e. through the iterative process. An incomplete cubic interpolation function is used for both the velocities and pressure within a triangular finite element. The discrete equations are integrated in time with backward finite differences. We illustrate the similarity between the (ψ,ζ) finite difference method and the ($\text{u}\text{,p}$) finite element method by calculations on the driven square cavity problem.     

A finite element method approach for the mechanobiological modeling of the osseointegration of a dental implant

The aim of this paper is to introduce a new mathematical model using a mechanobiological approach describing the process of osseointegration at the bone-dental implant interface in terms of biological and mechanical factors and the implant surface. The model has been computationally implemented by using the finite element method. The results show the spatial-temporal patterns distribution at the bone-dental implant interface and demonstrate the ability of the model to reproduce features of the wound healing process such as blood clotting, osteogenic cell migration, granulation tissue formation, collagen-like matrix displacements and new osteoid formation. The model might be used as a methodological basis for designing a dental tool useful to predict the degree of osseointegration of dental implants and subsequent formulation of mathematical models associated with different types of bone injuries and different types of implantable devices.     

A symmetric finite element formulation for the inverse problem of aquifer transmissivity

This paper presents a symmetric isoparametric finite element formulation for the inverse problem of aquifer transmissivity calculation with known piezometric head. An important aspect of the present formulation is that the groundwater flow equation describing the aquifer behavior is transformed into a second-order differential equation by introducing an artificial variable φ. The two-dimensional, line and transition elements derived based on the weak formulation of this transformed equation possess symmetric matrices. In the formulation of the line elements φ and its derivative in η direction are retained as primary variables. This permits modelling of sudden changes in aquifer width. The transition elements provide a natural connecting link between the two-dimensional elements and the line elements. The line elements provide an efficient means of modelling aquifers with unidirectional flow. Numerical examples are given. A comparison of the results obtained here with the Galerkin finite element solution (nonsymmetric formulation) clearly demonstrates the superiority of the formulation presented here.     

Asymptotic and finite element approximations for heat transfer in rotating compressible flow over an infinite porous disk

The laminar boundary layer equations for the compressible flow due to the finite difference in rotation and temperature rates are solved for the case of uniform suction through the disk. The effects of viscous dissipation on the incompressible flow are taken into account for any rotation rate, whereas for a compressible fluid they are considered only for a disk rotating in a stationary fluid. For the general case, the governing equations are solved numerically using a standard finite element scheme. Series solutions are developed for those cases where the suction effect is dominant. Based on the above analytical and numerical solutions, a new asymptotic finite element scheme is presented. By using this scheme one can significantly improve the pointwise accuracy of the standard finite element scheme.     

A stabilized finite element method for modeling of gas discharges

An efficient stabilized finite element method for modeling of gas discharge plasmas is represented which provides wiggle-free solutions without introducing much artificial diffusion. The stabilization is achieved by modifying the standard Galerkin test functions by means of a weighted quadratic term that results in a consistent Petrov-Galerkin formulation of the charge carriers in the plasma. Using the example of a glow discharge plasma in argon, it is shown that this efficient method provides more accurate results on the same spatial grid than the widely used finite difference approach proposed by Scharfetter-Gummel if the weighting factor is determined in dependence on the local Péclet number and the modified test functions are consistently applied to all terms of the governing equations.     

A stabilized mixed finite element method for Darcy flow

We develop new stabilized mixed finite element methods for Darcy flow. Stability and an a priori error estimate in the “stability norm” are established. A wide variety of convergent finite elements present themselves, unlike the classical Galerkin formulation which requires highly specialized elements. An interesting feature of the formulation is that there are no mesh-dependent parameters. Numerical tests confirm the theoretical results.     

Anisotropic finite element modeling for patient-specific mandible

This paper presents an ad hoc modular software tool to quasi-automatically generate patient-specific three-dimensional (3D) finite element (FE) model of the human mandible. The main task is taking into account the complex geometry of the individual mandible, as well as the inherent highly anisotropic material law. At first, by computed tomography data (CT), the individual geometry of the complete range of mandible was well reproduced, also the separation between cortical and cancellous bone. Then, taking advantage of the inherent shape nature as ‘curve’ long bone, the algorithm employed a pair of B-spline curves running along the entire upper and lower mandible borders as auxiliary baselines, whose directions are also compatible with that of the trajectory of maximum material stiffness throughout the cortical bone of the mandible. And under the guidance of this pair of auxiliary baselines, a sequence of B-spline surfaces were interpolated adaptively as curve cross-sections to cut the original geometry. Following, based on the produced curve contours and the corresponding curve cross-section surfaces, quite well structured FE volume meshes were constructed, as well as the inherent trajectory vector fields of the anisotropic material (orthotropic for cortical bone and transversely isotropic for cancellous bone). Finally, a sensitivity analysis comprising various 3D FE simulations was carried out to reveal the relevance of elastic anisotropy for the load carrying behavior of the mandible.     

Intelligent finite element method: An evolutionary approach to constitutive modeling

In this paper, a new approach is presented, for constitutive modeling of materials in finite element analysis, with potential applications in different engineering disciplines. The proposed approach provides a unified framework for modeling of complex materials, using evolutionary polynomial regression-based constitutive model (EPRCM), integrated in finite element analysis. Evolutionary polynomial regression (EPR) is a computing technique that generates a transparent and structured representation of the system being studied. The main advantage of EPRCM over conventional constitutive models is that it provides the optimum structure for the material constitutive model representation, as well as its parameters, directly from raw experimental (or field) data. The proposed algorithm provides a transparent relationship for the constitutive material model that can readily be incorporated in a finite element model (FEM). The incorporation of EPRCM into FEM will be presented and the application of the resulting methodology for material modeling in finite element analysis will be illustrated through two examples.     

A variational Germano approach for stabilized finite element methods

In this paper the recently introduced Variational Germano procedure is revisited. The procedure is explained using commutativity diagrams. A general Germano identity for all types of discretizations is derived. This relation is similar to the Variational Germano identity, but is not restricted to variational numerical methods. Based on the general Germano identity an alternative algorithm, in the context of stabilized methods, is proposed. This partitioned algorithm consists of distinct building blocks. Several options for these building blocks are presented and analyzed and their performance is tested using a stabilized finite element formulation for the convection–diffusion equation. Non-homogenous boundary conditions are shown to pose a serious problem for the dissipation method. This is not the case for the least-squares method although here the issue of basis dependence occurs. The latter can be circumvented by minimizing a dual-norm of the weak relation instead of the Euclidean norm of the discrete residual.     

An object-oriented blackboard based approach for automated finite element modeling and analysis of multichip modules

This paper addresses the application of the blackboard system architecture and object-oriented data abstraction techniques to the domain of finite element modeling and analysis. Specifically, a hierarchical object-oriented database was used to represent the physical system at different levels of abstraction including the user-defined physical system level, a computer-generated, simplified physical model level, and the finite element model level. Object link relationships within a given abstraction level and across different abstraction levels resulted in seamless bidirectional information exchange. The blackboard system architecture employed provided a framework for distributed cooperative problem solving, for the application of simplifying domain-specific modeling assumptions, and for integrating the various software modules that are involved in the entire finite element modeling and analysis process. These methodologies were implemented in a prototype computational tool calledIMCMA theIntelligentMultichipModuleAnalyzer. An example illustrates howIMCMA automates finite element thermal analysis of small integrated circuit features in multichip modules through a two-step finite element submodeling process.     

Computational modeling of electrochemical coupling: A novel finite element approach towards ionic models for cardiac electrophysiology

We propose a novel, efficient finite element solution technique to simulate the electrochemical response of excitable cardiac tissue. We apply a global–local split in which the membrane potential of the electrical problem is introduced globally as a nodal degree of freedom, while the state variables of the chemical problem are treated locally as internal variables on the integration point level. This particular discretization is efficient and highly modular since different cardiac cell models can be incorporated in a straightforward way through only minor local modifications on the constitutive level. Here, we derive the underlying algorithmic framework for a recently proposed ionic model for human ventricular cardiomyocytes, and demonstrate its integration into an existing nonlinear finite element infrastructure. To ensure unconditional algorithmic stability, we apply an implicit backward Euler scheme to discretize the evolution equations for both the electrical potential and the chemical state variables in time. To increase robustness and guarantee optimal quadratic convergence, we suggest an incremental iterative Newton–Raphson scheme and illustrate the consistent linearization of the weak form of the excitation problem. This particular solution strategy allows us to apply an adaptive time stepping scheme, which automatically generates small time steps during the rapid upstroke, and large time steps during the plateau, the repolarization, and the resting phases. We demonstrate that solving an entire cardiac cycle for a real patient-specific geometry characterized through a transmembrane potential, four ion concentrations, thirteen gating variables, and fifteen ionic currents requires computation times of less than ten minutes on a standard desktop computer.     

A quadrilateral finite difference plate element for nonlinear transient analysis of panels

A quadrilateral plate element for the analysis of nonlinear transient response of panels has been developed based on the variational finite difference method for an irregular mesh. Due to the superior computational characteristics of the variational finite difference method with a lesser degree of continuity constraint on the interpolation functions and the use of lower-order polynomials allowing faster numerical integration methods to be implemented, this plate element is quite competitive or perhaps even superior when compared with the conforming finite elements. Three illustrative problems have been solved using this plate element to demonstrate its capability and accuracy in analyzing the large deformation response of panels subject to dynamic loadings.Very favorable correlation was observed between analysis and experiment on large deformations of elasticplastic rectangular plates subject to intensive impulsive loadings. Similar correlation was also observed for circular plates modeled with this quadrilateral plate element under impulsive loadings. Finally, the large dynamic deformations of composite rectangular panels of graphite-epoxy, boron-epoxy, glass-epoxy, and isotropic material were analyzed and found in good agreement with other analytical results.     

A hybrid expert system for finite element modeling of fuselage frames

A hybrid expert system which integrates expert system with neural networks is developed for finite element modeling of fuselage frame of aircraft structure. Importance order parameters are introduced to quantify the modeling control. Expert knowledge of importance order parameters, node setting and element selection in fuselage frame modeling is presented. A neural network is employed to structure type classification. The modeling procedures of fuselage frame can be carried out automatically and efficiently by this system. Example shows the frame 1022 of MD-82 passenger aircraft which is modeled by this system automatically and successfully.     

Unconditionally stable concurrent procedures for transient finite element analysis

A new class of algorithms for transient finite element analysis which is amenable to an efficient implementation in parallel computers is proposed. The suitability of the method for parallel computation stems from the fact that, given an arbitrary partition of the finite element mesh, each subdomain in the partition can be processed over a time step independently and simultaneously with the rest. Both element-by-element and coarse partitions of the mesh are discussed. For the former, the proposed algorithms are shown to have the structure of an explicit scheme. In particular, no global equation solving effort is involved in the update procedure. However, in contrast to explicit schemes the proposed algorithms are shown to be unconditionally stable over a certain range of the algorithmic parameters. In structural dynamics problems, good accuracy is obtained with a constant time step integration. For heat conduction problems accuracy limitations suggest the use of a step-changing technique. When this is done, numerical tests indicate the good behavior of the method. The case in which the mesh is partitioned into a small number of subdomains, typically as many as processors in the computer, is also explored in detail. Good accuracy is obtained over a wide range of time steps. Finally, extensions to second- and higher-order accuracy methods are discussed.