Mesh Accuracy Research Articles

Reduction of mesh distortion effects for nine-node elements using corrected shape functions

The paper concerns two-dimensional nine-node quadrilateral elements based on the Green strain and the two-level approximations of strains. These approximations reduce locking well for regular meshes but cannot prevent the drop of accuracy when the side and central nodes are shifted from the middle positions.To reduce the deterioration of accuracy when nodes are shifted, we assess the corrected shape functions of Celia and Gray (1984 [10]) as a replacement for the standard isoparametric ones. In Celia and Gray (1984 [10]), the corrected shape functions were tested for an eight-node element, the heat conduction equation and the 4×4 integration. Here, we test their applicability to nine-node elements for plane elasticity and the 3×3-point integration.We modify and examine four elements: QUAD9⁎⁎ (Huang and Hinton, 1986 [15]), MITC9 [1] and ours 9-AS (Panasz and Wisniewski, 2008 [21]) and MITC9i (Wisniewski and Panasz, 2012 [26]). The elements are subjected to a range of tests involving several types of mesh distortions, to confirm passing of various forms of the patch test, to prove the absence of locking as well as to establish their coarse mesh accuracy and sensitivity to mesh distortions. We show that all the tested elements benefit from using the corrected shape functions, but still remain significant differences in their performance.

Two improvements in formulation of nine‐node element MITC9

SUMMARYThe paper concerns a well‐known two‐dimensional nine‐node quadrilateral element MITC9, which is based on two‐level approximations of strains (assumed strain method). The element has good accuracy, but does not pass the patch test.As the first improvement, we propose a modification of the element's transformations, partly resolving the problem with the patch test. The source of the problem is the use of covariant components in a (local) natural co‐basis, different at each sampling point.As the second improvement, we use the corrected shape functions of Celia MA, Gray WG. An improved isoparametric transformation for finite element analysis. International Journal for Numerical Methods in Engineering 1984; 20:1447–1459, extending their applicability to the nine‐node element for plane elasticity and the 3 × 3 integration. Originally, they are tested for an eight‐node element for the heat conduction equation and the 4 × 4 integration.The improved element, designated as MITC9i, is based on the Green strain and derived from the potential energy for the plane stress condition. It is subjected to a range of tests, to confirm that it passes the patch test for several types of mesh distortions, to prove its coarse mesh accuracy and the absence of locking as well as to establish its sensitivity to mesh distortions.The improved element MITC9i performs substantially better than the MITC9 element, QUAD9** element, and our previous 9‐AS element.Copyright © 2012 John Wiley & Sons, Ltd.

Numerical implementation of local effects due to two-dimensional discontinuous loads using special elements based on boundary integrals

Abstract In this paper, special-purpose elements are developed for solving local effects caused by discontinuous loads such as concentrated forces, line loads and patch loads applied in plane elastic structures. During the derivation of the special-purpose elements, the interior displacement and stress fields are composed of two parts: (1) the homogeneous solution part, which is represented by a linear combination of fundamental solutions at a number of source points outside the element domain; and (2) the particular load-dependent part, which is analytically represented by suitable local solutions. Meanwhile the independent frame displacements defined over the element boundary are approximated by conventional shape functions. The linkage between the two independent fields is established through use of a newly constructed hybrid variational functional, in which discontinuous loads are treated as generalized body forces. Using the property of delta function, the domain integral associated with discontinuous loads in the variational functional can be removed. The advantage of such special-purpose elements is that a large element, independent of the location of discontinuous loads, can be used to avoid the requirement of mesh refinement in the vicinity of the area with local loads. Numerical experiments are carried out to verify the special-purpose elements and to investigate their effectiveness in terms of mesh reduction and accuracy.

Automated subject-specific, hexahedral mesh generation via image registration

Generating subject-specific, all-hexahedral meshes for finite element analysis continues to be of significant interest in biomechanical research communities. To date, most automated methods “morph” an existing atlas mesh to match with a subject anatomy, which usually result in degradation in mesh quality because of mesh distortion. We present an automated meshing technique that produces satisfactory mesh quality and accuracy without mesh repair. An atlas mesh is first developed using a script. A subject-specific mesh is generated with the same script after transforming the geometry into the atlas space following rigid image registration, and is transformed back into the subject space. By meshing the brain in 11 subjects, we demonstrate that the technique's performance is satisfactory in terms of both mesh quality (99.5% of elements had a scaled Jacobian >0.6 while <0.01% were between 0 and 0.2) and accuracy (average distance between mesh boundary and geometrical surface was 0.07 mm while <1% greater than 0.5 mm). The combined computational cost for image registration and meshing was <4 min. Our results suggest that the technique is effective for generating subject-specific, all-hexahedral meshes and that it may be useful for meshing a variety of anatomical structures across different biomechanical research fields.

Spallation of Concrete under Dynamic Loading: Mesh Size Effect

Spallation of concrete under dynamic loading has been the hot issue of concern about civil engineering structures and protective engineering. In the present paper, the principle of propagation of stress wave and the induced spallation process along a one-dimensional bar and the RFPA-Dynamics code which considers the heterogeneity of the concrete materials are briefly introduced. In order to numerically investigate the effect of mesh size of numerical model on the computational results, the failure process of concrete with three mesh sizes under dynamic loads was numerically simulated using RFPA-Dynamics code. Numerical simulations show that the failure patterns of concrete with different mesh sizes under dynamic stress waves are different. It is found that for the numerical models with the same mechanical parameters, the smaller mesh size of the numerical model is, the longer propagation of stress wave delay, the smaller the compressive stress is, thus the greater the tensile stress is, and the number of cracks is increasing, in general, rupture is more serious. The accuracy of mesh is little to improve when the number of the grid increases to a certain value.

An efficient block parallel AMR method for two phase interfacial flow simulations

The direct numerical simulation of two phase interfacial flows can be computationally challenging, as the strong resolution needed to follow the deformations of the interface leads to a lot of time spent solving the whole computation domain. Efficient solution of such problems requires an adaptive mesh refinement capability to concentrate computational effort where it is most needed. In this paper a parallel adaptive algorithm to solve incompressible two-phase flows with surface tension is presented: the AMR is handled with the help of the PARAMESH package. The free interface between fluids is tracked via Level Set approach; the jump conditions at the interface for pressure and velocity are imposed by the Ghost Fluid method. A multigrid preconditioned BiCG-stab solver adapted to the AMR data structure has been developed to allow high density ratio computations (up to 1:1000). Special treatment has been done at the refinement jumps to maintain the fine mesh accuracy. Computational results are compared in different test cases with analytical solutions or literature, and show very good agreement with the references. The effectiveness of PARAMESH parallelization has been quite well maintained, as shown in the strong and weak scaling tests. Speed-up capabilities of the AMR are demonstrated.

An Adaptive Energy Mesh Constructor for Multigroup Library Generation for Transport Codes

Users’ demands for multigroup transport calculations are wide and diverse, encompassing routine, rough, and fast calculations as well as very precise simulations. For these reasons, the use of accurate and efficient multigroup cross-section libraries is needed. In this work, we present an adaptive energy mesh constructor (AEMC) that builds a multigroup mesh from predefined requisites of precision and calculation time. For a given self-shielding model and number of groups, AEMC looks for the optimal bounds of a multigroup mesh that minimizes the errors of the multigroup transport solutions for a predefined set of infinite homogeneous medium problems. We have applied this methodology to define two energy meshes for fast sodium reactor applications: a 600-group mesh associated with an extension of the Livolant-Jeanpierre self-shielding method and a 1200-group mesh based on subgroup self-shielding. Tests in homogeneous media prove that the multigroup solutions are almost equivalent to Monte Carlo simulations. Simplified one-dimensional transport calculations confirm the accuracy of the 1200-group mesh and show that this mesh provides a precision similar to that obtained with the well-validated 1968-group ECCO mesh. The same tests reveal that the 600-group mesh optimized for subgroup self-shielding offers a good compromise between simulation time and precision.

Adaptive Macro Finite Elements for the Numerical Solution of Monodomain Equations in Cardiac Electrophysiology

Many problems in biology and engineering are governed by anisotropic reaction-diffusion equations with a very rapidly varying reaction term. This usually implies the use of very fine meshes and small time steps in order to accurately capture the propagating wave while avoiding the appearance of spurious oscillations in the wave front. This work develops a family of macro finite elements amenable for solving anisotropic reaction-diffusion equations with stiff reactive terms. The developed elements are incorporated on a semi-implicit algorithm based on operator splitting that includes adaptive time stepping for handling the stiff reactive term. A linear system is solved on each time step to update the transmembrane potential, whereas the remaining ordinary differential equations are solved uncoupled. The method allows solving the linear system on a coarser mesh thanks to the static condensation of the internal degrees of freedom (DOF) of the macroelements while maintaining the accuracy of the finer mesh. The method and algorithm have been implemented in parallel. The accuracy of the method has been tested on two- and three-dimensional examples demonstrating excellent behavior when compared to standard linear elements. The better performance and scalability of different macro finite elements against standard finite elements have been demonstrated in the simulation of a human heart and a heterogeneous two-dimensional problem with reentrant activity. Results have shown a reduction of up to four times in computational cost for the macro finite elements with respect to equivalent (same number of DOF) standard linear finite elements as well as good scalability properties.

Surface reconstruction of torsos with and without scoliosis

Visible surface asymmetries such as uneven shoulders, waist and hips, shoulder height differences and a shoulder blade prominence are often the most troublesome features associated with adolescent scoliosis. Treatment considerations are influenced by the severity and changes over time of these asymmetries. Outcomes are judged on how well the asymmetries are improved towards a normal trunk shape. In this paper, a deformable self organizing feature map (SOFM) is used as a geometric surface reconstruction tool to model the torso surface of subjects with and without scoliosis. The proposed parameterization technique provides a means of quantifying the surface asymmetries and assessing the changes due to either natural history or the effects of treatment. For evaluation 10 control subjects without scoliosis and 10 adolescents with scoliosis were scanned and their torsos were reconstructed. This preliminary study demonstrates that in around 5 min a torso scan with 60,000 data points can be transformed into a 2562 nodes mesh using SOFM. The accuracy of the final mesh is around 1.40 mm on average. The high accuracy and speed of this technique, makes it well suitable to be used in a clinical setting to assess surface features of subjects with scoliosis.

Adaptive triangular mesh coarsening with centroidal Voronoi tessellations

We present a novel algorithm for adaptive triangular mesh coarsening. The algorithm has two stages. First, the input triangular mesh is refined by iteratively applying the adaptive subdivision operator that performs a so-called red-green split. Second, the refined mesh is simplified by a clustering algorithm based on centroidal Voronoi tessellations (CVTs). The accuracy and good quality of the output triangular mesh are achieved by combining adaptive subdivision and the CVTs technique. Test results showed the mesh coarsening scheme to be robust and effective. Examples are shown that validate the method.

Perspective on the geometric conservation law and finite element methods for ALE simulations of incompressible flow

This paper takes a fresh look at the geometric conservation law (GCL) from the perspective of the finite element method (FEM) for incompressible flows. The GCL arises naturally in the context of Arbitrary Lagrangian Eulerian (ALE) formulations for solving problems on deforming domains. GCL compliance is traditionally interpreted as a consistency criterion for applying an unsteady flow solution algorithm to simulate exactly a uniform flow on a deforming domain. We introduce an additional requirement: the time integrator must maintain its fixed mesh accuracy when applied to deforming meshes. A review of the literature shows that while many authors use an ALE FEM, few of them discuss the GCL issues. We show how a fixed mesh unsteady FEM using high order time integrator (up to fifth order in time) can be transposed to solve problems on deforming meshes and preserve its fixed mesh high order temporal accuracy. An appropriate construction of the divergence of the mesh velocity guarantees GCL compliance while a separate construction of the mesh velocity itself allows the time-integrator to deliver its fixed mesh high order temporal accuracy on deforming domains. Analytical error analysis of problems with closed form solutions provides insight on the behavior of the time integrators. It also explains why high order temporal accuracy is achieved with a conservative formulation of the incompressible Navier–Stokes equations, while only first order time accuracy is observed with the non-conservative formulation and all time-integrators investigated here. We present thorough time-step and grid refinement studies for simple problems with closed form solutions and for a manufactured solution with a non-trivial flow on a deforming mesh. In all cases studied, the proposed reconstructions of the mesh velocity and its divergence for the conservative formulation lead to optimal time accuracy on deforming grids.

Nine-node shell elements with 6 dofs/node based on two-level approximations. Part I: Theory and linear tests

The paper concerns 9-node quadrilateral shell elements derived for Reissner's kinematics. They are based on the Green strain and potential energy, and are applicable to large (unrestricted) rotations. The characteristic features of the developed elements are as follows: 1. Drilling rotation is included via the drill rotation constraint (RC) imposed by the penalty method. Hence, the elements have 6 dofs per node, i.e. three displacements and three rotational parameters, including drilling rotation. 2. Transverse shear and membrane locking as well as the in-plane shear over-stiffening are avoided using the two-level approximation applied to the strain (assumed strain method). This method does not affect the drilling RC. 3. A modification of the two-level approximation method is proposed, consisting in treating the sampling and the numerical integration together, which results in six sampling points being replaced by two sampling lines. The two-level approximation is applied to components in the ortho-normal basis at the element center, which differs our element from the MITC family of elements, which uses the covariant strain components. 4. Selective reduced integration (SRI) approach is revised. The total functional is split into several parts, and a suitable integration rule is found for each part, yielding an efficient element which shows very good mesh convergence. Two 9-node shell elements are developed and subjected to a range of benchmark tests, to establish the sensitivity to mesh distortion, the coarse mesh accuracy, and to confirm the lack of locking. Our results are compared with results obtained by the MITC9 element of ADINA and the S9R5 element of ABAQUS.

Curvature Computation for Triangular Meshes based on Local Parametrization

AbstractA local parametrization based curvature computation technique for triangular meshes is presented. The computation process starts with interpolating the region-of-interest of the given mesh with a Loop subdivision surface. The interpolation technique guarantees that the resulting surface reflects the local shape of the mesh, including features such as edges and corners; no data simplification is necessary. A blending technique is then applied to vicinities of extra-ordinary vertices to ensure continuity and boundedness of curvature at each extra-ordinary vertex. This blending process does not change the value of the surface at the extra-ordinary points. Curvatures for each given point are subsequently computed based on standard parametrization for Loop surfaces. Advantages of the new technique include that curvature can be computed in any direction for any point of the given mesh and higher accuracy of the computed results due to precise representation obtained by the interpolation process. Test re...

An efficient and accurate short-characteristics solver for radiative transfer problems

In Turek (Computing 54, 27–38, 1995), the concept of the generalized mean intensity has been proposed as a special numerical approach to the (linear) radiative transfer equations which can result in a significant reduction of the dimension of the discretized system, without eliminating any information for the specific intensities. Moreover, in combination with Krylov-space methods (CG, Bi-CGSTAB, etc.), robust and very efficient solvers as extensions of the classical approximate Λ-iteration have been developed. In this paper, the key tool is the combination of special renumbering techniques together with finite difference-like discretization strategies for the arising transport operators which are based on short-characteristic upwinding techniques of variable order and which can be applied to highly unstructured meshes with locally varying mesh widths. We demonstrate how such special upwinding schemes can be constructed of first order, and particularly of second order accuracy, always leading to lower triangular system matrices on general meshes. As a consequence, the global matrix assembling can be avoided (“on-the-fly”) so that the storage cost is almost optimal, and the solution of the corresponding convection-reaction subproblems for each direction can be obtained very efficiently. As a further consequence, this approach results in a direct solver in the case of no scattering, while in the case of non-vanishing absorption and scattering coefficients the resulting convergence rates for the full systems depend only on their ratio and the absolute size of these physical quantities, but only weakly on the grid size or mesh topology. We demonstrate these results via prototypical configurations and we examine the resulting accuracy and efficiency for different computational domains, meshes and problem parameters.

A comparison of rotation-free triangular shell elements for unstructured meshes

Many engineering applications require accurate and rapidly computed thin-shell elements. Rotation-free (RF) shell elements include the bending behaviour of thin shells without introducing any additional degrees of freedom compared to a membrane element. Instead, constant curvatures are approximated from the out-of-plane displacements of a patch of usually four triangular elements. A consequence of this is that the accuracy for irregular meshes has been unsatisfactory. The aim of this study is to find an RF shell element which is accurate also for unstructured meshes. The main difference between existing elements is whether they assume two-dimensional constant curvatures over the patch or use superposition of one-dimensional constant curvatures for the three pairs of triangles. The first assumption fulfils constant curvatures for a Kirchhoff plate exactly, whereas the second and most common assumption only approximates constant curvatures. The first assumption is significantly more resistant to element shape distortions, whereas the second assumption is slightly faster to compute and more appropriate on boundaries where one or more elements are missing or several neighbouring elements share a side. The combination is significantly more accurate for irregular meshes than other comparable RF elements for linear benchmark tests.

ニューラルネットワークを用いた防撓パネルの変位及び応力の予測

At the design stage of the large-scale steel structure, displacement and stress are generally analyzed by using the FEM. At the initial design stage, it is required for evaluating conditions of numerous design candidates. Therefore, it is not efficient to use FEM with fine mesh for analyzing the strength of structures in spite of requiring accuracy in the evaluation. In this paper, Neural Network System is developed for predicting displacement and stress based on a coarse mesh with the same accuracy of fine mesh. It is able to predict accurate displacement and stress at stiffened plate using FEM analytical results with coarse mesh as input factors for Neural Network System. In order to improve the approximation accuracy of Neural Network System, Multiple linear regression method is introduced for selecting input factors for a Neural Network System. This technique improves the accuracy of Neural Network System and reduce manpower used for structure design. The proposed Neural Network System is able to predict displacement and stress in spite of changing the load, plate thickness, and the number of stiffeners.

An efficient adaptive multigrid algorithm for predicting thin film flow on surfaces containing localised topographic features

Gravity-driven continuous thin film flow over a plane, containing well-defined single and grouped topographic features, is modelled as Stokes flow using lubrication theory. The associated time-dependent, nonlinear, coupled set of governing equations is solved using a full approximation storage (FAS) multigrid algorithm by employing automatic mesh adaptivity, the power, efficiency and accuracy of which is demonstrated by comparing the results with corresponding global fine-mesh solutions. These show that automatic grid refinement effectively restricts the use of fine grids to regions of rapid flow development which, for the topographies considered, includes the topography itself, the upstream Capillary ridge, downstream surge region, and the characteristic bow wave. It is shown that for the accurate solution of the associated lubrication equations, adaptive multigrid offers increased flexibility together with a significant reduction in memory requirement. This is further demonstrated by solving the problem of transient flow over a trench topography, generated by a sinusoidally varying inlet condition.

Simple and accurate four‐node axisymmetric element

AbstractBased on the assumed strain method, a simple four‐node axisymmetric solid element is introduced. The assumed strain field is carefully selected to preserve the correct rank of the element stiffness matrix and to achieve high accuracy. The strain field is developed in conjunction with orthogonal projections and no matrix inversions are needed. The coarse mesh accuracy in bending and in typical axisymmetric load cases is excellent even for nearly incompressible materials. The strain‐driven format obtained is well suited for materials with non‐linear stress–strain relations. Several numerical examples are presented where the excellent performance of the proposed simple element is verified. Copyright © 2006 John Wiley &amp; Sons, Ltd.

Polymer Melt Flow in Plane Channels: Effects of the Viscous Dissipation and Axial Heat Conduction

ABSTRACT The plastic transforming industry has shown considerable growth in the last years. In this context, commercial simulators have been developed, some of which combine simplified mathematical models with rheological properties of commercial polymers. In spite of the successes, these approximations are not able to capture important details of the flow behavior. The present work addresses some aspects of the polymer melt flow in plane channels using a more elaborate mathematical formulation based on the full momentum and energy conservation laws. The physical equations are discretized using finite differences based on a collocated mesh and second-order spatial accuracy formulas. Solutions featuring the development of hydrodynamic and thermal boundary layers are presented for a commercial polymer.

A Stabilized Mixed Finite Element Method for Nearly Incompressible Elasticity

Abstract We present a new multiscale/stabilized finite element method for compressible and incompressible elasticity. The multiscale method arises from a decomposition of the displacement field into coarse (resolved) and fine (unresolved) scales. The resulting stabilized-mixed form consistently represents the fine computational scales in the solution and thus possesses higher coarse mesh accuracy. The ensuing finite element formulation allows arbitrary combinations of interpolation functions for the displacement and stress fields. Specifically, equal order interpolations that are easy to implement but violate the celebrated Babushka-Brezzi inf-sup condition, become stable and convergent. Since the proposed framework is based on sound variational foundations, it provides a basis for a priori error analysis of the system. Numerical simulations pass various element patch tests and confirm optimal convergence in the norms considered.