Level Of Mesh Refinement Research Articles

Large Eddy Simulation of Stable Supersonic Jet Impinging on Flat Plate

This paper describes a numerical study based on large eddy simulation of the flow induced by a supersonic jet impinging on a flat plate in a stable regime. This flow involves very high velocities, shocks, and intense shear layers. Performing large eddy simulation on such flows remains a challenge because of the shock discontinuities. Here, large eddy simulation is performed with an explicit third-order compressible solver using a unstructured mesh, a centered scheme, and the Smagorinsky model. Three levels of mesh refinement (from 7 to 22 million cells) are compared in terms of instantaneous and averaged flowfields (shock and recirculation zone positions), averaged flow velocity and pressure fields, wall pressure, root mean square pressure fields, and spectral content using one and two-point analyses. The effects of numerical dissipation and turbulent viscosity are compared on the three grids and shown to be well controlled. The comparison of large eddy simulation with experimental data shows that the finest grid (a 22 million cell mesh) ensures grid-independent results not only for the mean and rms fields but also for higher statistics such as single and two-point correlation functions.

Investigation of allowable time-step sizes for generalized finite element analysis of the transient heat equation

This paper investigates the heat equation for domains subjected to an internal source with a sharp spatial gradient. The solution is first approximated using linear finite elements, and sufficiently small time-step sizes to yield stable simulations. The main area of interest is then in the ability to approximate the solution using Generalized Finite Elements, and again explore the time-step limitations required for stable simulations. Both high order elements, as well as elements with special enrichments are used to generate solutions. When compared to linear finite elements, the high order elements deliver better accuracy at a given level of mesh refinement, but do not offer an increase in critical time-step size. When special enrichment functions are used, the solution can be approximated accurately on very coarse meshes, while yielding solutions which are both accurate and computationally efficient. The major conclusion of interest is that the significantly larger element size yields larger allowable time-step sizes while still maintaining stability of the time-stepping algorithm.

Numerical and Experimental Models of the Mandible

This study aimed to validate a numerical model of an intact mandible for further development of a new TMJ implant. Numerical and experimental models of the biomechanics of the mandible were elaborated to characterize the human temporomandibular joint and to approach the development of a condyle implant. The model of the mandible was obtained through the use of a polymeric replica of a human cadaveric mandible and through 3D geometry acquisition. The three-dimensional finite element model was generated as a tetrahedral finite element mesh. The level of mesh refinement was established via a convergence test and a model with more than 50,000 degrees of freedom was required to obtain analysis accuracy. The functional loading cases included muscle loading in four different load boundary conditions. The same boundary conditions were applied to the experimental model. The strains were measured with an experimental procedure using electric resistance strain gauges applied on the external surface of the mandible. The mechanical response is shown and discussed in terms of strains, principal numerical and measured strains. This study proved that FE models of the mandible can reproduce experimental strains within an overall agreement of 10%. The FE models correctly reproduced bone strains under different load configurations and therefore can be used for the design of a novel TMJ implant considering other load configurations and bone mechanical properties.

Extension of the analytic nodal diffusion solver ANDES to triangular-Z geometry and coupling with COBRA-IIIc for hexagonal core analysis

In this paper the extension of the multigroup nodal diffusion code ANDES, based on the Analytic Coarse Mesh Finite Difference (ACMFD) method, from Cartesian to hexagonal geometry is presented, as well as its coupling with the thermal–hydraulic (TH) code COBRA-IIIc for hexagonal core analysis. In extending the ACMFD method to hexagonal assemblies, triangular-Z nodes are used. In the radial plane, a direct transverse integration procedure is applied along the three directions that are orthogonal to the triangle interfaces. The triangular nodalization avoids the singularities, that appear when applying transverse integration to hexagonal nodes, and allows the advantage of the mesh subdivision capabilities implicit within that geometry. As for the thermal–hydraulics, the extension of the coupling scheme to hexagonal geometry has been performed with the capability to model the core using either assembly-wise channels (hexagonal mesh) or a higher refinement with six channels per fuel assembly (triangular mesh). Achieving this level of TH mesh refinement with COBRA-IIIc code provides a better estimation of the in-core 3D flow distribution, improving the TH core modelling. The neutronics and thermal–hydraulics coupled code, ANDES/COBRA-IIIc, previously verified in Cartesian geometry core analysis, can also be applied now to full three-dimensional VVER core problems, as well as to other thermal and fast hexagonal core designs. Verification results are provided, corresponding to the different cases of the OECD/NEA-NSC VVER-1000 Coolant Transient Benchmarks.

Development of an object-oriented finite element program with adaptive mesh refinement for multi-physics applications

In this paper, an object-oriented framework for numerical analysis of multi-physics applications is presented. The framework is divided into several basic sets of classes that enable the code segments to be built according to the type of problem to be solved. Fortran 2003 was used in the development of this finite element program due to its advantages for scientific and engineering programming and its new object-oriented features. The program was developed with h-type adaptive mesh refinement, and it was tested for several classical cases involving heat transfer, fluid mechanics and structural mechanics. The test cases show that the adaptive mesh is refined only in the localization region where the feature gradient is relatively high. The overall mesh refinement and the h-adaptive mesh refinement were justified with respect to the computational accuracy and the CPU time cost. Both methods can improve the computational accuracy with the refinement of mesh. The overall mesh refinement causes the CPU time cost to greatly increase as the mesh is refined. However, the CPU time cost does not increase very much with the increase of the level of h-adaptive mesh refinement. The CPU time cost can be saved by up to 90%, especially for the simulated system with a large number of elements and nodes.

Electromyographic study of postural adaptation during mandibular advancement

This work introduces a novel method of automating the process of patient-specific finite element (FE) model development using a mapped mesh technique. The objective is to map a predefined mesh (template) of high quality directly onto a new bony surface (target) definition, thereby yielding a similar mesh with minimal user interaction. To bring the template mesh into correspondence with the target surface, a deformable registration technique based on the FE method has been adopted. The procedure has been made hierarchical allowing several levels of mesh refinement to be used, thus reducing the time required to achieve a solution. Our initial efforts have focused on the phalanx bones of the human hand. Mesh quality metrics, such as element volume and distortion were evaluated. Furthermore, the distance between the target surface and the final mapped mesh were measured. The results have satisfactorily proven the applicability of the proposed method.

Optimisation of brace treatment with numerical modelling

This work introduces a novel method of automating the process of patient-specific finite element (FE) model development using a mapped mesh technique. The objective is to map a predefined mesh (template) of high quality directly onto a new bony surface (target) definition, thereby yielding a similar mesh with minimal user interaction. To bring the template mesh into correspondence with the target surface, a deformable registration technique based on the FE method has been adopted. The procedure has been made hierarchical allowing several levels of mesh refinement to be used, thus reducing the time required to achieve a solution. Our initial efforts have focused on the phalanx bones of the human hand. Mesh quality metrics, such as element volume and distortion were evaluated. Furthermore, the distance between the target surface and the final mapped mesh were measured. The results have satisfactorily proven the applicability of the proposed method.

ParTopS: compact topological framework for parallel fragmentation simulations

Cohesive models are used for simulation of fracture, branching and fragmentation phenomena at various scales. Those models require high levels of mesh refinement at the crack tip region so that nonlinear behavior can be captured and physical results obtained. This imposes the use of large meshes that usually result in computational and memory costs prohibitively expensive for a single traditional workstation. If an extrinsic cohesive model is to be used, support for dynamic insertion of cohesive elements is also required. This paper proposes a topological framework for supporting parallel adaptive fragmentation simulations that provides operations for dynamic insertion of cohesive elements, in a uniform way, for both two- and three-dimensional unstructured meshes. Cohesive elements are truly represented and are treated like any other regular element. The framework is built as an extension of a compact adjacency-based serial topological data structure, which can natively handle the representation of cohesive elements. Symmetrical modifications of duplicated entities are used to reduce the communication of topological changes among mesh partitions and also to avoid the use of locks. The correctness and efficiency of the proposed framework are demonstrated by a series of arbitrary insertions of cohesive elements into some sample meshes.

Automated hexahedral meshing of anatomic structures using deformable registration

This work introduces a novel method of automating the process of patient-specific finite element (FE) model development using a mapped mesh technique. The objective is to map a predefined mesh (template) of high quality directly onto a new bony surface (target) definition, thereby yielding a similar mesh with minimal user interaction. To bring the template mesh into correspondence with the target surface, a deformable registration technique based on the FE method has been adopted. The procedure has been made hierarchical allowing several levels of mesh refinement to be used, thus reducing the time required to achieve a solution. Our initial efforts have focused on the phalanx bones of the human hand. Mesh quality metrics, such as element volume and distortion were evaluated. Furthermore, the distance between the target surface and the final mapped mesh were measured. The results have satisfactorily proven the applicability of the proposed method.

On the convergence of the panel method for potential problems with non-smooth domains

The linear convergence rate of the panel method is often observed in problems with smooth boundaries. When corners are present in the problem domain, the convergence rate slows down greatly. A study on the convergence rate of the panel method for potential problems with non-smooth domains was conducted and is reported in this paper. It has been found that for Dirichlet boundary value problems, the relative errors in the normal flux produced by the panel method at corners persist regardless of mesh refinement level. This is the main factor that deteriorates the overall convergence rate of the panel method for Dirichlet type of problems with non-smooth domains. Methods to improve convergence and accuracy at corners are proposed and are demonstrated in the paper.

Multilevel algorithms for Rannacher–Turek finite element approximation of 3D elliptic problems

Generalizing the approach of a previous work of the authors, dealing with two-dimensional (2D) problems, we present multilevel preconditioners for three-dimensional (3D) elliptic problems discretized by a family of Rannacher Turek non-conforming finite elements. Preconditioners based on various multilevel extensions of two-level finite element methods (FEM) lead to iterative methods which often have an optimal order computational complexity with respect to the number of degrees of freedom of the system. Such methods were first presented by Axelsson and Vassilevski in the late-1980s, and are based on (recursive) two-level splittings of the finite element space. An important point to make is that in the case of non-conforming elements the finite element spaces corresponding to two successive levels of mesh refinement are not nested in general. To handle this, a proper two-level basis is required to enable us to fit the general framework for the construction of two-level preconditioners for conforming finite elements and to generalize the method to the multilevel case. In the present paper new estimates of the constant γ in the strengthened Cauchy–Bunyakowski–Schwarz (CBS) inequality are derived that allow an efficient multilevel extension of the related two-level preconditioners. Representative numerical tests well illustrate the optimal complexity of the resulting iterative solver, also for the case of non-smooth coefficients. The second important achievement concerns the experimental study of AMLI solvers applied to the case of micro finite element (μFEM) simulation. Here the coefficient jumps are resolved on the finest mesh only and therefore the classical CBS inequality based convergence theory is not directly applicable. The obtained results, however, demonstrate the efficiency of the proposed algorithms in this case also, as is illustrated by an example of microstructure analysis of bones.

Multilevel preconditioning of rotated bilinear non-conforming FEM problems

Preconditioners based on various multilevel extensions of two-level finite element methods (FEM) lead to iterative methods which often have an optimal order computational complexity with respect to the number of degrees of freedom of the system. Such methods were first presented in [O. Axelsson, P.S. Vassilevski, Algebraic multilevel preconditioning methods I, Numer. Math. 56 (1989) 157–177; O. Axelsson, P.S. Vassilevski, Algebraic multilevel preconditioning methods II, SIAM J. Numer. Anal. 27 (1990) 1569–1590] on (recursive) two-level splittings of the finite element space. The key role in the derivation of optimal convergence rate estimates is played by the constant γ in the so-called strengthened Cauchy–Bunyakowski–Schwarz (CBS) inequality, associated with the angle between the two subspaces of the splitting. More precisely, the value of the upper bound for γ ∈ ( 0 , 1 ) is a part of the construction of various multilevel extensions of the related two-level methods. In this paper algebraic two-level and multilevel preconditioning algorithms for second-order elliptic boundary value problems are constructed, where the discretization is done using Rannacher–Turek non-conforming rotated bilinear finite elements on quadrilaterals. An important point to make is that in this case the finite element spaces corresponding to two successive levels of mesh refinement are not nested in general. To handle this, a proper two-level basis is required to enable us to fit the general framework for the construction of two-level preconditioners for conforming finite elements and to generalize the method to the multilevel case. The proposed variants of the hierarchical two-level basis are first introduced in a rather general setting. Then, the parameters involved are studied and optimized. The major contribution of the paper is the derived estimates of the constant γ in the strengthened CBS inequality which is shown to allow the efficient multilevel extension of the related two-level preconditioners. Representative numerical tests well illustrate the optimal complexity of the resulting iterative solver.

Uncertainty quantification for chaotic fluid dynamics

AbstractWe study mesh convergence issues for a representative chaotic flow problem. Convergence in the sense of pointwise convergence fails as new structures emerge with each new level of mesh refinement. We formulate convergence in terms of statistical averages. Care is made to divide the problem into homogeneous regions, and statistical convergence is found for each such region. Automated feature extraction is the tool used for this decomposition.We study of averaged equations, and the closure problem which arises in their derivation. The accuracy of the closure terms is discussed. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A hierarchical method for discrete structural topology design problems with local stress and displacement constraints

AbstractIn this paper, we present a hierarchical optimization method for finding feasible true 0–1 solutions to finite‐element‐based topology design problems. The topology design problems are initially modelled as non‐convex mixed 0–1 programs.The hierarchical optimization method is applied to the problem of minimizing the weight of a structure subject to displacement and local design‐dependent stress constraints. The method iteratively treats a sequence of problems of increasing size of the same type as the original problem. The problems are defined on a design mesh which is initially coarse and then successively refined as needed. At each level of design mesh refinement, a neighbourhood optimization method is used to treat the problem considered.The non‐convex topology design problems are equivalently reformulated as convex all‐quadratic mixed 0–1 programs. This reformulation enables the use of methods from global optimization, which have only recently become available, for solving the problems in the sequence. Numerical examples of topology design problems of continuum structures with local stress and displacement constraints are presented. Copyright © 2006 John Wiley & Sons, Ltd.

A comparison of two and three-dimensional analyses of fatigue crack closure

Plasticity-induced fatigue crack closure is an important mechanism in the reduction of the effective stress intensity factor range for a fatigue crack. A calculation of the level of reduction would allow more accurate predictions of fatigue crack growth rate. However, modelling plasticity-induced closure is not straightforward, particularly when the three-dimensional aspects of the problem are included. Some simplification is possible by reducing the problem to two dimensions, but it is not always clear how this can be achieved for practical crack geometries. In this work, two-dimensional plane stress and plane strain finite element analyses are used to predict crack opening in a centre-cracked plate. The results of these analyses are compared with those of a plane stress strip yield analysis and those of a three-dimensional finite element analysis. Results are obtained for different R-ratios and stress levels. Reasonable agreement is found between the plane stress finite element and strip yield results for higher levels of applied stress levels where an excessively high level of mesh refinement is not required. Plane stress finite element crack opening results agree with three-dimensional finite element results for the surface and plane strain finite element results agree with three-dimensional finite element results for the mid-thickness. The implications of the results for the behaviour of three-dimensional cracks are discussed.

Finite element modeling of crystal plasticity with grains shaped as truncated octahedrons

Different modeling strategies are tested for the prediction of texture development and microscopic strain heterogeneity in cold-rolled ULC steel and in multiphase steel under uniaxial tension. The polycrystalline aggregate is represented by a finite element mesh that is loaded under periodic boundary conditions. Grains are shaped as cubes or as truncated octahedrons, defining three levels of mesh refinement. Simulations rely on a simplified implementation of crystal plasticity, in which elastic strains are considered infinitesimal. The constitutive law is integrated fully implicitly in a reference frame bounded to the crystal lattice. Accuracy of the time-integration procedure is assessed by referring to some predictions of the crystal plasticity algorithm developed by Kalidindi et al. [Crystallographic texture evolution in bulk deformation processing of FCC metals. J. Mech. Phys. Solid 40 (1992) 537–569]. Then, results of the micro–macro modeling are compared to experimental data. It is found that the simulations with truncated octahedral grains yield improved predictions compared to those with cuboidal grains.

A new element-by-element method for trajectory calculations with tetrahedral finite element meshes

A new method to track massless particles in a three-dimensional flow field is presented. The method is based on an element-by-element approach coupled with a predictor–corrector shooting scheme and does not use any time step. By analogy with time-dependent schemes, the number of shootings is related to an equivalent number of time steps. The method has been implemented in a finite element framework using unstructured tetrahedral finite element meshes. However, it is general enough so that it can be implemented in finite difference and finite volume frameworks as well. It has been tested on a variety of flow systems namely: the Poiseuille flow in an empty circular pipe, the rotating flow in a stirred tank, the shear flow in a square tank and the flow through a static mixer. Accuracy has been found to depend on the accuracy of the velocity computation, the number of points per element and the level of mesh refinement. Copyright © 2006 John Wiley & Sons, Ltd.

Error comparison in tracked anduntracked spherical simulations

This paper follows our earlier work on axisymmetric flows [1–4] where algorithms, theories, experiments, simulations, applications, and validations were presented. Here we study the effectiveness and efficiency of explicit front tracking by comparing the L 1-error for spherical shock refraction simulations with and without tracking. We find that front tracking reduces the level of mesh refinement needed to achieve a specified error tolerance by a significant factor compared to corresponding methods without tracking, thus substantially reducing the computational time as well as memory usage for simulations with contacts or material interfaces.

Spatially Averaged Local Strains in Textile Composites Via the Binary Model Formulation

A previously published computational model of textile composites known as the Binary Model is generalized to allow systematic study of the effects of mesh refinement. Calculations using different meshing orders show that predictions of local strains are mesh independent when the strains are averaged over gauge volumes whose dimensions are greater than or equal to approximately half the width dimensions of a single tow. Strains averaged over such gauges are favored for use in failure criteria for predicting various mechanisms of failure in a textile composite, including transverse cracking within tows, kink band failure in compression, tensile tow rupture, and shear failure. For the highest order representations (infinitely dense meshes), the generalized formulation of the Binary Model necessarily approaches conventional finite element meshing strategies for textile composites in its predictions. However, the work reported here implies that usefully accurate predictions of spatially averaged strains can be obtained even at the lowest level of mesh refinement. This preserves great simplicity in the model set-up and rapid computation for relatively large features of structural components. Calculations for some textile structures provide insight into the strength or relative absence of textile effects in local strains for different loading configurations.

The use of graded finite elements in the study of elastic wave propagation in continuously nonhomogeneous materials

Finite elements with graded properties are used to simulate elastic wave propagation in functionally graded materials. The graded elements are formulated with continuously nonhomogeneous material property fields and compared to conventionally formulated homogeneous elements. An example problem is solved for the two-dimensional case to show the potential benefits of the graded formulation. It is observed that the conventional elements give a discontinuous stress field in the direction perpendicular to the material property gradient, while the graded elements give a continuous distribution. In the one-dimensional case, the solutions are compared to the analytical solutions by Chiu and Erdogan [J. Sound Vib. 222 (3) (1999) 453]. The results show that for identical levels of mesh refinement, the graded formulation produces similar spatial resolution, and temporal resolution for the range of boundary value problems studied. Observations are explained and their implications for numerical modeling are discussed.