Adaptive Finite Element Approach Research Articles

The Role of the Assembly Force in the Tribocorrosion Behaviour of Hip Implant Head-Neck Junctions: An Adaptive Finite Element Approach.

The cyclic loading, in the corrosive medium of the human body, results in tribocorrosion at the interface of the head-neck taper junction of hip implants. The resulting metal ions and wear debris adversely affect the local tissues. The force applied by surgeons to assemble the junction has proven to play a major role in the mechanics of the taper junction which, in turn, can influence the tribocorrosion damage. Recently, finite element method has been used to predict the material loss at the head-neck interface. However, in most finite element studies, the contribution of electrochemical corrosion has been ignored. Therefore, a detailed study to investigate the influence of the assembly force on the tribocorrosive behaviour of the head-neck junction, which considers both the mechanical and chemical material removal, is of paramount interest. In this study, a finite-element-based algorithm was used to investigate the effect of assembly force on the tribocorrosion damage at the junction interface, for over four million cycles of simulated level gait. The patterns of the material removal in the modelling results were compared with the damage patterns observed in a group of retrieved modular hip implants. The results of this study showed that for different cases, chemical wear was in the range of 25-50% of the total material loss, after four million cycles. A minimum assembly force (4 kN for the studied cases) was needed to maintain the interlock in the junction. The computational model was able to predict the damage pattern at the retrieved head-neck interface.

3-D Marine CSEM Modeling in General Anisotropic Media by Using an Adaptive Finite Element Approach Based on the Vector-Scalar Potential

3-D Marine CSEM Modeling in General Anisotropic Media by Using an Adaptive Finite Element Approach Based on the Vector-Scalar Potential

A two-dimensional, adaptive finite element approach for simulation of backward erosion piping

Backward erosion piping (BEP) is a type of internal erosion that significantly contributes to internal erosion being a leading cause of levee and dam failures, second only to overtopping. Despite its significance, very few numerical or analytical tools exist for assessing the progression of backward erosion pipes. Further, existing tools do not automatically refine the mesh near the pipe tip, nor do they automatically search for the weakest erosion path in the foundation. Both features are critical for assessing piping when conditions along the pipe path vary due to natural geologic variability or engineered features such as filter trenches and partial cutoff walls. In the present study, a two-dimensional, auto-adaptive, finite element program has been developed that solves the coupled groundwater flow and erosion pipe flow equations in a piecewise, steady-state manner to simulate BEP progression using automated, local mesh refinement. Criteria for pipe advancement and pipe widening are presented that permit pipe growth through the mesh along the path of least resistance, and an approximation to account for foundation depth is introduced. Examples are presented in which BEP progression is simulated beneath a levee to illustrate the analysis method.

A goal-oriented dual-weighted adaptive finite element approach for the optimal control of a nonsmooth Cahn–Hilliard–Navier–Stokes system

This paper is concerned with the development and implementation of an adaptive solution algorithm for the optimal control of a time-discrete Cahn–Hilliard–Navier–Stokes system with variable densities. The free energy density associated with the Cahn–Hilliard system incorporates the double-obstacle potential which yields an optimal control problem for a family of coupled systems in each time instant of a variational inequality of fourth order and the Navier–Stokes equation. A dual-weighted residual approach for goal-oriented adaptive finite elements is presented which is based on the concept of C-stationarity. The overall error representation depends on primal residuals weighted by approximate dual quantities and vice versa as well as various complementarity mismatch errors. Details of the numerical realization of the adaptive concept and a report on the numerical tests are given.

Adaptive Multilevel Monte Carlo Methods for Stochastic Variational Inequalities

While multilevel Monte Carlo (MLMC) methods for the numerical approximation of partial differential equations with random coefficients enjoy great popularity, combinations with spatial adaptivity seem to be rare. We present an adaptive MLMC finite element approach based on deterministic adaptive mesh refinement for the arising “pathwise” problems and outline a convergence theory in terms of desired accuracy and required computational cost. Our theoretical and heuristic reasoning together with the efficiency of our new approach are confirmed by numerical experiments.

Fully adaptive higher-order finite element analysis of fast transient phenomena on overhead lines

An alternative fully adaptive finite element approach is used for modeling fast transient phenomena on overhead lines. The methodology is based on rewriting the hyperbolic transmission equation for voltage to two specific equations where one unknown is the voltage itself and another unknown is its partial derivative with respect to time. The system is then solved by the higher-order finite element method in space and by the backward differential formula in time. Computational tests carried out on several typical examples prove that the approach exhibits numerous advantages compared with finite difference time-domain method.

Real-Time Error Control for Surgical Simulation.

To present the first a posteriori error-driven adaptive finite element approach for real-time simulation, and to demonstrate the method on a needle insertion problem. We use corotational elasticity and a frictional needle/tissue interaction model. The problem is solved using finite elements within SOFA.1 For simulating soft tissue deformation, the refinement strategy relies upon a hexahedron-based finite element method, combined with a posteriori error estimation driven local -refinement. We control the local and global error level in the mechanical fields (e.g., displacement or stresses) during the simulation. We show the convergence of the algorithm on academic examples, and demonstrate its practical usability on a percutaneous procedure involving needle insertion in a liver. For the latter case, we compare the force-displacement curves obtained from the proposed adaptive algorithm with that obtained from a uniform refinement approach. Error control guarantees that a tolerable error level is not exceeded during the simulations. Local mesh refinement accelerates simulations. Our work provides a first step to discriminate between discretization error and modeling error by providing a robust quantification of discretization error during simulations.

Adaptive finite element modeling of marine controlled-source electromagnetic fields in two-dimensional general anisotropic media

In this paper, we extend the scope of numerical simulations of marine controlled-source electromagnetic (CSEM) fields in a particular case of anisotropy (dipping anisotropy) to the general case of anisotropy by using an adaptive finite element approach. In comparison to a dipping anisotropy case, the first order spatial derivatives of the strike-parallel components arise in the partial differential equations for generally anisotropic media, which cause a non-symmetric linear system of equations for finite element modeling. The adaptive finite element method is employed to obtain numerical solutions on a sequence of refined unstructured triangular meshes, which allows for arbitrary model geometries including bathymetry and dipping layers. Numerical results of a 2D anisotropic model show both anisotropy strike and dipping angles have great influence on the marine CSEM responses.

An adaptive finite element approach for neutron transport equation

In this paper, we develop an adaptive element refinement strategy that progressively refines the elements in appropriate regions of domain to solve even-parity Boltzmann transport equation. A posteriori error approach has been used for checking the approximation solutions for various sizes of elements. The local balance of neutrons in elements is utilized as an error assessment. To implement the adaptive approach a new neutron transport code FEMPT, finite element modeling of particle transport, for arbitrary geometry has been developed. This code is based on even-parity spherical harmonics and finite element method. A variational formulation is implemented for the even-parity neutron transport equation for the general case of anisotropic scattering and sources. High order spherical harmonic functions expansion for angle and finite element method in space is used as trial function. This code can be used to solve the multi-group neutron transport equation in highly complex X– Y geometries with arbitrary boundary condition. Due to powerful element generator tools of FEMPT, the description of desired and complicated 2D geometry becomes quite convenient. The numerical results show that the locally adaptive element refinement approach enhances the accuracy of solution in comparison with uniform meshing approach.

Adaptive Finite Element Simulation of the Time-dependent Simplified P N Equations

The steady-state simplified P N approximation to the radiative transfer equation has been successfully applied to many problems involving radiation. Recently, time-dependent simplified P N equations have been derived by an asymptotic analysis similar to the asymptotic derivation of the steady-state SP N equations (Frank et al. in J. Comput. Phys. 226:2289–2305, 2007). In this paper, we present computational results for the time-dependent SP N equations in two dimensions, obtained by using an adaptive finite element approach. Several numerical comparisons with other existing models are shown.

Adaptive finite element method for simulation of optical nano structures

AbstractWe discuss realization, properties and performance of the adaptive finite element approach to the design of nano‐photonic components. Central issues are the construction of vectorial finite elements and the embedding of bounded components into the unbounded and possibly heterogeneous exterior. We apply the finite element method to the optimization of the design of a hollow core photonic crystal fiber. Thereby we look at the convergence of the method and discuss automatic and adaptive grid refinement and the performance of higher order elements. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Adaptive unstructured grid finite element simulation of two-dimensional magnetotelluric fields for arbitrary surface and seafloor topography

We present an adaptive unstructured triangular grid finite element approach for effectively simulating plane-wave diffusive electromagnetic fields in 2-D conductivity structures. The most striking advantage of irregular grids is their potential to incorporate arbitrary geometries including surface and seafloor topography. Adaptive mesh refinement strategies using an a posteriori error estimator yield most efficient numerical solutions since meshes are only refined where required. We demonstrate the robustness of this approach by comparison with analytical solutions and previously published numerical simulations. Maximum errors may systematically be reduced to, for example, 0.8 per cent for the apparent resistivity and 0.2° in the phase. An additional accuracy study of the thickness of the air layer in E-polarization suggests to keep a minimum thickness depending on lateral conductivity contrasts within the earth. Furthermore, we point out the new quality and flexibility of our simulation technique by addressing two marine magnetotelluric applications. In the first case, we discuss topographic effects associated with a synthetic sinusoidal sea bottom model and in the second case, we show a close-to-reality scenario using real bathymetry data from the East Pacific Rise at 17°S.

Adaptive Finite Element Methods in Electrochemistry

In this article, we review some of our previous work that considers the general problem of numerical simulation of the currents at microelectrodes using an adaptive finite element approach. Microelectrodes typically consist of an electrode embedded (or recessed) in an insulating material. For all such electrodes, numerical simulation is made difficult by the presence of a boundary singularity at the electrode edge (where the electrode meets the insulator), manifested by the large increase in the current density at this point, often referred to as the edge effect. Our approach to overcoming this problem has involved the derivation of an a posteriori bound on the error in the numerical approximation for the current that can be used to drive an adaptive mesh-generation algorithm, allowing calculation of the quantity of interest (the current) to within a prescribed tolerance. We illustrate the generic applicability of the approach by considering a broad range of steady-state applications of the technique.

Adaptive finite element methodology for tumour angiogenesis modelling

AbstractIn this paper, we consider an adaptive finite element approach for reliable, efficient solution of a class of continuum models for tumour‐induced angiogenesis. The ideas are demonstrated using an established three equation reaction/transport model that simulates aspects of tumour‐induced angiogenesis in a deterministic manner. The weak variational formulation and finite element approximation scheme for the model are developed, and a statistical approach for concurrent adaptive mesh refinement and coarsening is described. The appropriate form of the model and solution dependence on choice of parameters are explored. Computational results are presented for 1D, 2D and 3D geometry models. The effectiveness of the open‐source, parallel adaptive software library (LibMesh) that is being developed in the CFDLab at the University of Texas is also demonstrated. Copyright © 2006 John Wiley & Sons, Ltd.

Formula omitted] for microlithography applications: Modeling, simulation and optimization of the crystal growth process

Optimization is the ultimate goal of numerical modeling of crystal growth processes. We present the most important physical phenomena to establish a numerical process model for an industrial vertical gradient freeze (VGF) process of barium fluoride ( BaF 2 ). After reviewing the governing equations we describe an adaptive finite element approach for a phase change problem with internal radiation. We describe an interface to a commercial software package for industrial crystal growth processes to define a numerical process model in a highly automatized way. In this way we can apply an external optimization algorithm to our process model. We use a modified quasi-Newton method to propose an improvement of an industrial VGF growth process of BaF 2 .

A First-Order System Least Squares Finite Element Method for the Shallow Water Equations

A least squares finite element method for the first-order system of the shallow water equations is proposed and studied. The method combines a characteristic-based time discretization with a least squares finite element approach which approximates the water level and the velocity field in H1 and H(div), respectively. The linearized least squares functional is shown to be elliptic, uniformly as the time-step size $\tau$ approaches zero, with respect to a suitably weighted norm. Moreover, Lipschitz continuity of the Fréchet derivative is shown with respect to this norm. This implies that the local evaluation of the nonlinear least squares functional constitutes an a posteriori error estimator on which an adaptive refinement technique may be based. The efficiency of such an adaptive finite element approach is tested numerically for a test problem involving the surface flow in a widening channel leading to a recirculating velocity field.

Fast and Adaptive Finite Element Approach for Modeling Brain Shift

Objective: In this paper we introduce a finite element-based strategy for simulation of brain deformation occurring during neurosurgery. The phenomenon, known as brain shift, causes a decrease in the accuracy of neuronavigation systems that rely on preoperatively acquired data. This can be compensated for with a computational model of the brain deformation process. By applying model calculations to preoperative images, an update within the operating room can be performed.Methods: One of the crucial concerns in the context of developing a physical-based model is the choice of governing equations describing the physics of the phenomenon. In this work, deformation of brain tissue is expressed in terms of a 3D consolidation model for a linearly elastic and porous fluid. The next crucial issue is ensuring stable calculations within the chosen model. For this purpose, we developed a special technique for generating the underlying geometry for the simulation. With this technique an unstructured grid consisting of regular tetrahedra is created, whereupon time-dependent finite element simulation is performed in an adaptive manner.Results: We applied our algorithm to preoperative MR scans and investigated the value of the method. Due to the adaptivity of the method, only 5-10% of the computing time was needed as compared to traditional finite element approaches based on a uniformly subdivided grid. The results of the experiments were compared to the corresponding intraoperative MR scans. A close match between the computed deformation of the brain and the displacement resulting from the intraoperative data was observed.Conclusion: A model-based approach for the simulation of brain shift is presented. In this computational model the brain tissue is described as an elastic and porous material using Biot consolidation theory. Validating experiments conducted with MR data provided promising results.

An adaptive finite element approach for a mixed Reissner–Mindlin plate formulation

We use a recently introduced mixed finite element method for the Reissner–Mindlin plate model and study its numerical properties in practical use. After introducing a technique for stabilizing the discretisation, we derive an a posteriori error estimate and study the question of reliability. Numerical examples prove that it enables an efficient adaptive scheme.

An adaptive finite element approach for frictionless contact problems

The derivation of an a posteriori error estimator for frictionless contact problems under the hypotheses of linear elastic behaviour and infinitesimal deformation is presented. The approximated solution of this problem is obtained by using the finite element method. A penalization or augmented-Lagrangian technique is used to deal with the unilateral boundary condition over the contact boundary. An a posteriori error estimator suitable for adaptive mesh refinement in this problem is proposed, together with its mathematical justification. Up to the present time, this mathematical proof is restricted to the penalization approach. Several numerical results are reported in order to corroborate the applicability of this estimator and to compare it with other a posteriori error estimators. Copyright © 2001 John Wiley & Sons, Ltd.

Adaptative finite element simulation of currents at microelectrodes to a guaranteed accuracy. Application to a simple model problem

In this series of papers we consider the general problem of numerical simulation of the currents at microelectrodes using an adaptive finite element approach. Microelectrodes typically consist of an electrode embedded (or recessed) in an insulating material. For all such electrodes, numerical simulation is made difficult by the presence of a boundary singularity at the electrode edge (where the electrode meets the insulator), manifested by the large increase in the current density at this point, often referred to as the ‘edge-effect’. Our approach to overcoming this problem involves the derivation of an a posteriori bound on the error in the numerical approximation for the current that can be used to drive an adaptive mesh-generation algorithm. This allows us to calculate the current to within a prescribed tolerance. We begin by demonstrating the power of the method for a simple model problem — an E reaction mechanism at a microdisc electrode — for which the analytical solution is known. In this paper we give the background to the problem, and show how an a posteriori error bound can be used to drive an adaptive mesh-generation algorithm. We then use the algorithm to solve our model problem and obtain very accurate results on comparatively coarse meshes in minimal computing time. We give the technical details of the background theory and the derivation of the error bound in the accompanying paper.