Level Of Mesh Refinement Research Articles

Finite element analysis of plasticity-induced fatigue crack closure: an overview

Finite element analysis is perhaps the most commonly used numerical method to model plasticity-induced fatigue crack closure. The state-of-the-art is reviewed and a comprehensive overview is presented, summarizing issues which must be considered and emphasizing potential difficulties. These include mesh refinement level, crack advancement schemes, crack shape evolution, geometry effects, and crack opening value assessment techniques.

Prevention of mesh-dependent damage growth in finite element simulations of crack formation in acrylic bone cement

Peak stress levels predicted in finite element analysis (FEA) usually depend on mesh density, due to singular points in the model. In an earlier study, an FEA algorithm was developed to simulate the damage accumulation process in the cement mantle around total hip replacement (THR) implants. It allows cement crack formation to be predicted, as a function of the local cement stress levels. As the simulation is driven by mesh-dependent peak stresses, predicted crack formation rates are also likely to be mesh dependent. The aim of this study was to evaluate the mesh dependence of the predicted crack formation process, and to present a method to reduce the mesh dependence. Crack-propagation experiments were simulated. Experimental specimens, representing transverse slices of cemented THR reconstructions, were subjected to cyclic torsional loading. Crack development around the corners of the stem was monitored. The experiments were simulated using three meshes with increasing levels of mesh refinement. Crack locations and orientations were accurately predicted, and were virtually independent of the level of mesh refinement. However, the experimental crack propagation rates were overestimated considerably, increasing with mesh refinement. To eliminate the effect of stress singularities around the corners of the stem, a stress averaging algorithm was applied in the simulation. This algorithm redistributed the stresses by weighted spatial averaging. When damage accumulation was computed based on averaged stresses, the crack propagation rates predicted were independent of the level of mesh refinement. The critical distance, a parameter governing the effect of the averaging algorithm, was optimized such that the predicted crack propagation rates accurately corresponded to the experimental ones. These results are important for the validity and standardization of pre-clinical testing methods for orthopaedic implants.

Towards a reliable method for predicting the rheological properties of multiphase fluids

Measurement of the constitutive behavior of multiphase fluids is one of the most interesting and difficult problems in rheology. The boundary element method is particularly suited to modeling the flow of such fluids because of the relative ease of representing their complex geometry accurately without excessive meshing difficulties. Many papers have been published on the subject, but to the author's knowledge, the literature on the actual implementation of such methods is scarce. Here, a parallel implementation of a code based on the completed double layer boundary integral equation, with Barnes–Hut multipole acceleration, is described briefly. After determining the level of mesh refinement required to obtain convergent results, the code is used to predict the rheological behavior of suspensions of rigid particles of different shapes. The effect of concentration and spatial distributions on the rheological behavior of the suspensions is investigated. It is found that the properties of suspensions of rigid particles are correctly predicted for concentrations up to 30% solids, while at higher concentrations accurate predictions require excessively fine discretization. A new hybrid numerical-asymptotic technique is proposed, which produces a significant improvement in accuracy accompanied by a reduction in computational cost.

High order compact scheme with multigrid local mesh refinement procedure for convection diffusion problems

We derive a new fourth order compact finite difference scheme which allows different meshsize in different coordinate directions for the two-dimensional convection diffusion equation. A multilevel local mesh refinement strategy is used to deal with the local singularity problem. A corresponding multilevel multigrid method is designed to solve the resulting sparse linear system. Numerical experiments are conducted to show that the local mesh refinement strategy works well with the high order compact discretization scheme to recover high order accuracy for the computed solution. Our solution method is also shown to be effective and robust with respect to the level of mesh refinement and the anisotropy of the problems.

Towards a reliable method for predicting the rheological properties of multiphase fluids

Measurement of the constitutive behavior of multiphase fluids is one of the most interesting and difficult problems in rheology. The boundary element method is particularly suited to modeling the flow of such fluids because of the relative ease of representing their complex geometry accurately without excessive meshing difficulties. Many papers have been published on the subject, but to the author's knowledge, the literature on the actual implementation of such methods is scarce. Here, a parallel implementation of a code based on the completed double layer boundary integral equation, with Barnes–Hut multipole acceleration, is described briefly. After determining the level of mesh refinement required to obtain convergent results, the code is used to predict the rheological behavior of suspensions of rigid particles of different shapes. The effect of concentration and spatial distributions on the rheological behavior of the suspensions is investigated. It is found that the properties of suspensions of rigid particles are correctly predicted for concentrations up to 30% solids, while at higher concentrations accurate predictions require excessively fine discretization. A new hybrid numerical-asymptotic technique is proposed, which produces a significant improvement in accuracy accompanied by a reduction in computational cost.

A Weakly Overlapping Domain Decomposition Preconditioner for the Finite Element Solution of Elliptic Partial Differential Equations

We present a new two-level additive Schwarz domain decomposition preconditioner which is appropriate for use in the parallel finite element solution of elliptic partial differential equations (PDEs). As with most parallel domain decomposition methods each processor may be assigned one or more subdomains, and the preconditioner is such that the processors are able to solve their own subproblem(s) concurrently. The novel feature of the technique proposed here is that it requires just a single layer of overlap in the elements which make up each subdomain at each level of refinement, and it is shown that this amount of overlap is sufficient to yield an optimal preconditioner. Some numerical experiments---posed in both two and three space dimensions---are included to confirm that the condition number when using the new preconditioner is indeed independent of the level of mesh refinement on the test problems considered.

The flow through a plenum-runner system—a comparison of experiment and computation

This paper describes the comparison between experimental and computational results for the flow in an inlet system that contains a plenum. Here, the main focus is on the details of the computations and the comparison with experimental results for flow in the plenum. Details of the experiment are described elsewhere. By varying mesh density, boundary conditions, discretization schemes and turbulence models, a wide-ranging study of the accuracy of computational techniques for an industrial problem has been made. In particular, levels of mesh refinement for converged numerical solutions have been determined, together with performance levels for the various settings used. In terms of velocity prediction, when the flow is driven by boundary conditions specified at the outlet more accurate results are achieved. This is due to the flow at the inlet being calculated more accurately in these cases. The effects of different discretization schemes and turbulence models on the velocity predictions are small. Pressure predictions are, however, improved by more complex turbulence models such as the renormalization group (RNG) model.

EDICT for 3D computation of two-fluid interfaces

We present the 3D implementation and applications of the enhanced-discretization interface-capturing technique (EDICT) in computation of unsteady flows with two-fluid interfaces. In such computations, EDICT can be used as a very effective method, which combines the flexibility and efficiency of interface-capturing techniques with the accuracy provided by enhanced discretization at the interfaces. A stabilized finite element interface-capturing technique is used as the base formulation to solve, over a typically non-moving mesh, the Navier–Stokes equations and an advection equation governing the interface function. To increase the accuracy in modeling the interfaces, we use finite element functions with multiple components at and near the interfaces, with each component coming from a different level of mesh refinement. With its parallel implementation on advanced high-performance computing platforms such as the CRAY T3E, EDICT is a powerful tool for the simulation of a complex, 3D unsteady flow problems with two fluid-interfaces, including free surfaces.

Development and analysis of an adaptive transport scheme

An approach to solving the advection dominated atmospheric mass transport problem which adaptively constructs and updates the computational mesh is implemented and analyzed. The formulation of the mesh adaptation algorithm allows for information from other model processes as well as transport to influence mesh refinement. Comparisons to other methods are limited to a pure advection test problem. The scheme is based on a Petrov–Galerkin finite element method using quadratic interpolating polynomials over triangular elements. The temporal component of the transport equation is discretized using the Crank–Nicolson method. A single time step is applied to the entire domain regardless of mesh refinement levels. Two means of upwind biasing the solution via modification of the finite element weighting functions are investigated. Criteria for refining and unrefining the mesh based on advective flux and an estimate of the error are proposed and compared. The effects of varying the number of time steps taken between mesh refinements are analyzed. Results for adapted mesh solutions of a test problem are shown to be superior to corresponding uniform mesh finite element solutions. Comparisons to several other popular uniform mesh advection schemes indicate either equal or improved accuracy for the adapted mesh solutions.

Parallel computation of unsteady compressible flows with the EDICT

Recently, the Enhanced-Discretization Interface-Capturing Technique (EDICT) was introduced for simulation of unsteady flow problems with interfaces such as two-fluid and free-surface flows. The EDICT yields increased accuracy in representing the interface. Here we extend the EDICT to simulation of unsteady viscous compressible flows with boundary/shear layers and shock/expansion waves. The purpose is to increase the accuracy in selected regions of the computational domain. An error indicator is used to identify these regions that need enhanced discretization. Stabilized finite-element formulations are employed to solve the Navier-Stokes equations in their conservation law form. The finite element functions corresponding to enhanced discretization are designed to have two components, with each component coming from a different level of mesh refinement over the same computational domain. The primary component comes from a base mesh. A subset of the elements in this base mesh are identified for enhanced discretization by utilizing the error indicator. A secondary, more refined, mesh is constructed by patching together the second-level meshes generated over this subset of elements, and the second component of the functions comes from this mesh. The subset of elements in the base mesh that form the secondary mesh may change from one time level to other depending on the distribution of the error in the computations. Using a parallel implementation of this EDICT-based method, we apply it to test problems with shocks and boundary layers, and demonstrate that this method can be used very effectively to increase the accuracy of the base finite element formulation.

Análise da difusão durante a salga de queijos por elementos finitos: influência combinada da discretização tempo-espacial

Numerical simulation of solute (sodium chloride) transfer process during salting through a three-dimensional matrix (cheese) was studied applying the finite element method. It was verified that both mesh refinement level and time step length were relevant to control oscillatory behaviors even when unconditional stability schemes as Crank-Nicolson and modified Euler were used. A discussion of the combined influence of time and space adaptation in the context of diffusion problem is also presented, taking in consideration a lumped capacity matrix to overcome the difficulties and determine the minimum length of the time step. Differential mathematical modeling had as theoretical basis the Fick’s second law. The proposed model brought good estimation of salt gain in the soft cheese studied. Choosing the appropriate mesh and a convenient time step length we suggest Crank-Nicolson scheme for the simulation of diffusion during cheese brining.

Enhanced-Discretization Interface-Capturing Technique (EDICT) for computation of unsteady flows with interfaces

We present the Enhanced-Discretization Interface-Capturing Technique (EDICT) for computation of unsteady flow problems with interfaces, such as two-fluid and free-surface flows. In EDICT, we solve, over a non-moving mesh, the Navier-Stokes equations together with an advection equation governing the evolution of an interface function with two distinct values identifying the two fluids. The starting point for the spatial discretization of these equations are the stabilized finite element formulations which possess good stability and accuracy properties. To increase the accuracy in modeling the interfaces, we use finite element functions corresponding to enhanced discretization at and near the interface. These functions are designed to have multiple components, with each component coming from a different level of mesh refinement over the same computational domain. The primary component of the functions for velocity and pressure comes from the base mesh called Mesh-1. A subset of the elements in Mesh-1 are identified to be at or near the interface, and depending on where the interface is, this subset could change from one time level to another. A Mesh-2 is constructed by patching together the second-level meshes generated over this subset of elements, and the second component of the functions for velocity and pressure comes from Mesh-2. For the interface function, we have a third component coming from a Mesh-3 which is constructed by patching together the third-level meshes generated over a subset of elements in Mesh-2. With parallel computation of the test problems presented here, we demonstrate that the EDICT can be used very effectively to increase the accuracy of the base finite element formulations.

Modeling the biomechanics of the mandible: A three-dimensional finite element study

Three-dimensional finite element models of a partially edentulated human mandible were generated to calculate the mechanical response to simulated isometric biting and mastication loads. The level of mesh refinement was established via a convergence test and showed that a model with over 30,000 degrees of freedom was required to obtain analysis accuracy. The functional loading cases included muscle loading based on an algorithm that assigns muscle forces in accordance with muscle cross-sectional area, while maintaining static equilibrium. Results were found for isometric application of unilateral and bilateral bite and mastication loading, and two different sets of displacement boundary conditions were imposed at the condyles. The mechanical response is shown in terms of displacements, principal strains, and a new measure called the ‘mechanical intensity scalar’. For each load case studied, there was substantial bending in the molar region of the corpus and high tensile strains in the anterior portion of the ramus.

Solving knife-edge scattering problems using singular boundary elements

A multidomain boundary element method (BEM) is employed to study the scattering of time-harmonic acoustic waves by knife-edge structures in an acoustic field. Analysis shows that singularities exist in such structures, and that the behavior of the acoustic pressure and its derivative near a knife edge resembles that of the displacement and stress near a crack tip in an elastic body. Singular elements are employed to model the behavior on the part of the boundary in the vicinity of the singularity. The advantage of the singular elements is seen through several numerical examples involving mufflerlike structures containing baffles and one with a knife-edge scattering in full space. For a given level of mesh refinement, both the near-field and far-field solutions are improved with the introduction of singular elements.

Numerically induced chaos in the nonlinear Schrödinger equation.

The cubic nonlinear Schr\"odinger equation and some of its discretizations, one of which is integ- rable, are studied. Apart from the integrable version the discretizations produce chaotic solutions for intermediate levels of mesh (mode) refinement. Chaos disappears when the discretization is fine enough and convergence to a quasiperiodic solution is obtained. Details are given for finite difference calculations, although similar results are also obtained by Fourier spectral methods. Results regarding a forced nonlinear Schr\"odinger equation are briefly described.