Mesh Adaptation Strategy Research Articles

An Implicit FEM Solution of a PBE Model of Gibbsite Crystallization with Constant and Nonlinear Kinetics

Numerical solution for a 1-D dynamic population balance crystallization model that includes gibbsite secondary nucleation and crystal growth kinetics was developed. The implemented numerical algorithm combines an implicit Galerkin formulation of the finite element method (FEM) with Newton iterations, variable Gear-type time-step and adaptive nonuniform mesh strategies. The numerical solution of the crystallization model is compared to the analytical solution derived for the case of constant gibbsite crystallization kinetics. For this case, it is shown that the numerical solution was considerably stabilized with the introduction of the artificial diffusion term and reduction of the relative error of Newton’s iterative step. Furthermore, the numerical algorithm is tested for the case of nonlinear gibbsite crystallization kinetics demonstrating its ability to deliver consistent solutions for both nucleation and crystal growth dominant cases. In each of the cases considered, the model solution, valid for an i...

A mesh adaptation procedure for periodic domains

AbstractIn this paper, an original technique is developed in order to build adaptive meshes on periodic domains. The new approach has the important property that it is code‐reused. The procedure is used against three different algorithms, namely, MAdLib ( Int. J. Numer. Meth. Engng 2000; in press), mmg (Proc. 17th Int. Meshing Roundtable, 2008) and the couple Yams (Rapport Technique RT‐0252, 2001) /Ghs3d (Proc. 8th Int. Meshing Roundtable, 1999). None of the latter algorithms needs to be adapted before it is applied to periodic domains. Some examples of adaptation are presented based on analytical, isotropic and anisotropic mesh‐size fields. Periodicity in translation and rotation both are considered. Finally, the mesh adaptation strategy is used in order to reduce the computational cost of a prediction of strain heterogeneity throughout a periodic polycrystalline aggregate deforming by dislocation slip. Copyright © 2011 John Wiley & Sons, Ltd.

Adaptive Meshing of 2D Heterogeneous Objects Using Material Quadtree

AbstractIn finite element analysis (FEA), adaptive meshing of an object is usually preferred. With adaptive meshing, geometric accuracies of the mesh and more accurate FEA solution can be achieved while maintaining the computational efficiency. Numerous researches about adaptive meshing have been done and most of the existing schemes can generate meshes properly when the objects under meshing are homogeneously constituted. With the advent of heterogeneous objects, traditional adaptive meshing schemes become inadequate as the material heterogeneities of the mesh nodes and mesh elements are not taken into consideration. Inaccurate FEA results may result when a traditional adaptive mesh for a heterogeneous object is used as input geometry for the FEA. To cope with this problem, proper adaptive meshing schemes for heterogeneous objects should be developed. In this paper, the problem in 2D case is considered and a material quadtree is introduced. In this material quadtree, apart from the geometries of a hetero...

Simulation of Suction Caisson Penetration in Seabed Using an Adaptive Mesh Technique

This paper reports results of a numerical investigation into the suction caissons penetration in sand. A 3D finite element model is used for this purpose. The water saturated soil domain is modelled as a two-phase medium, consisting of soil skeleton and pore-water phases. Nonlinear behavior of the solid phase is described by means of a bounding-surface plasticity model. The caisson itself is modelled by solid elements and non-porous linear elastic properties. The soil-soil and soil-caisson interactions along the penetration path are modelled with a master/slave contact algorithm using contact surfaces and a frictional contact interface. A re-meshing technique is also used to avoid excessive distortion of the finite elements along the caisson-soil penetration path.The model was first substantiated against laboratory test data. The numerical model was found to present acceptable agreements with the experimental measurements. The verified model was then used to study the influence of parameters such as soil internal friction, penetration speed and adaptive mesh schemes on the installation behavior of suction caissons in sands.

An h-adaptive solution of the spherical blast wave problem

Shock waves and contact discontinuities usually appear in compressible flows, requiring a fine mesh in order to achieve an acceptable accuracy of the numerical solution. The usage of a mesh adaptation strategy is convenient as uniform refinement of the whole mesh becomes prohibitive in three-dimensional (3D) problems. An unsteady h-adaptive strategy for unstructured finite element meshes is introduced. Non-conformity of the refined mesh and a bounded decrease in the geometrical quality of the elements are some features of the refinement algorithm. A 3D extension of the well-known refinement constraint for 2D meshes is used to enforce a smooth size transition among neighbour elements with different levels of refinement. A density-based gradient indicator is used to track discontinuities. The solution procedure is partially parallelised, i.e. the inviscid flow equations are solved in parallel with a finite element SUPG formulation with shock capturing terms while the adaptation of the mesh is sequentially performed. Results are presented for a spherical blast wave driven by a point-like explosion with an initial pressure jump of 105 atmospheres. The adapted solution is compared to that computed on a fixed mesh. Also, the results provided by the theory of self-similar solutions are considered for the analysis. In this particular problem, adapting the mesh to the solution accounts for approximately 4% of the total simulation time and the refinement algorithm scales almost linearly with the size of the problem.

Multi‐scale modeling of moving interface problems with flux and field jumps: Application to oxidative degradation of ceramic matrix composites

AbstractProblems involving reaction and species diffusion involve field and flux jumps at a moving reaction front. In multi‐scale problems such as carbon fiber composite oxidation, these effects need to be tracked at the microscopic scale of individual carbon fibers. A multi‐scale model is derived in this paper for predicting species distribution in such problems using a fully coupled multi‐scale homogenization approach. The homogenized fluxes from the micro‐scale are derived using Hill's macro‐homogeneity condition accounting for both flux jumps and species density field jumps at the reacting interface in the micro‐scale unit cell. At the macro‐scale, the competition between the transport of reacting species (oxygen) and the reaction product (carbon dioxide) is modeled using homogenized mass conservation equations. The moving reaction front in carbon fibers at the micro‐scale is tracked using level set method and an adaptive meshing strategy. The macroscopic weight loss of the composite when exposed to oxygen is simulated as a function of time using a coupled finite element methodology at various locations in a validated macroscopic model. Copyright © 2010 John Wiley & Sons, Ltd.

Three dimensional adaptive meshing scheme applied to the control of the spatial representation of complex field pattern in electromagnetics

We present an improved adaptive mesh process based on Riemannian transformation to control the accuracy in high field gradient representation for diffraction problems. Such an adaptive meshing is applied in representing the electromagnetic intensity around a metallic submicronic spherical particle, which is known to present high gradients in limited zones of space including the interference pattern of the electromagnetic field. We show that, the precision of the field variation being controlled, this improved scheme permits drastically decreasing the computational time as well as the memory requirements by adapting the number and the position of nodes where the electromagnetic field must be computed and represented.

Numerical Simulation of Supersonic Nonpremixed Turbulent Combustion in a Scramjet Combustor Model

Nonpremixed turbulent combustion is studied in situations relevant to those encountered in advanced airbreathing engines operating at high flight Mach number values. Accurate simulations would require modeling both finite rate chemical kinetics and molecular diffusion effects (mixing) as well as the complex compressible flowfield structure associated with the presence of multiple shock and expansion waves that significantly influence combustion. The chemical kinetics are modeled by using a Lagrangian framework for turbulent combustion that has been recently extended to reactive high-speed flows. The corresponding framework incorporates both finite rate kinetics and turbulent molecular mixing rates with minimal additional parameters. An efficient anisotropic mesh adaptation strategy is used to obtain a satisfactory description of both nonreactive and reactive compressible flowfields. The present study confirmed that the proposed approach provides a well-suited framework for future developments devoted to nonpremixed turbulent combustion modeling in high-speed flows. A detailed comparison with data gathered from an experimental study of a laboratory scramjet model indicated the validity of the approach.

An augmented Lagrangian approach to Bingham fluid flows in a lid-driven square cavity with piecewise linear equal-order finite elements

We study an augmented Lagrangian approach to Bingham fluid flows in a lid-driven square cavity. The piecewise linear equal-order finite element spaces for both the velocity and the pressure approximations, proposed and analyzed by Latché and Vola [18], are applied. Based on the resulting regularity of the numerical solutions, a mesh adaptive strategy is proposed to render the yield surfaces of desired resolution. The corresponding numerical scheme is formulated for general Herschel–Bulkley models, and its validity is verified on the benchmark model: Bingham fluid flows in a lid-driven square cavity.

Dynamic analysis of laminated composite plates traversed by a moving mass based on a first-order theory

The dynamic response of angle-ply laminated composite plates traversed by a moving mass or a moving force is investigated. For this purpose, a finite element method based on the first-order shear deformation theory is used. Stationary and adaptive mesh techniques have been applied as two different meshing schemes. The adaptive mesh strategy is then used to avoid off-nodal position of moving mass. In this manner, the finite element mesh is continuously adapted to follow and comply with the path of moving mass. A Newmark direct integration method is employed to solve the equations of motion. Parametric study is directed to find out how different parameters like mass of the moving object as well as the type of the angle-ply laminated composite plates affect the dynamic response. Numerical results show the significant effects of the stacking order on the dynamic responses of the composite structures under a moving mass. It is found that although [30/-60/-60/30] lamination shows the highest maximum vertical deflection but [-45/45/451-45] lamination has the highest value of the dynamic amplification factor. The dynamic amplification factor for different stacking orders and mass velocities is less than 1.25. (C) 2010 Elsevier Ltd. All rights reserved.

A new anisotropic mesh adaptation method based upon hierarchical a posteriori error estimates

A new anisotropic mesh adaptation strategy for finite element solution of elliptic differential equations is presented. It generates anisotropic adaptive meshes as quasi-uniform ones in some metric space, with the metric tensor being computed based on hierarchical a posteriori error estimates. A global hierarchical error estimate is employed in this study to obtain reliable directional information of the solution. Instead of solving the global error problem exactly, which is costly in general, we solve it iteratively using the symmetric Gauss--Seidel (GS) method. Numerical results show that a few GS iterations are sufficient for obtaining a reasonably good approximation to the error for use in anisotropic mesh adaptation. The new method is compared with several strategies using local error estimators or recovered Hessians. Numerical results are presented for a selection of test examples and a mathematical model for heat conduction in a thermal battery with large orthotropic jumps in the material coefficients.

3D phase-field simulations of interfacial dynamics in Newtonian and viscoelastic fluids

This work presents a three-dimensional finite-element algorithm, based on the phase-field model, for computing interfacial flows of Newtonian and complex fluids. A 3D adaptive meshing scheme produces fine grid covering the interface and coarse mesh in the bulk. It is key to accurate resolution of the interface at manageable computational costs. The coupled Navier–Stokes and Cahn–Hilliard equations, plus the constitutive equation for non-Newtonian fluids, are solved using second-order implicit time stepping. Within each time step, Newton iteration is used to handle the nonlinearity, and the linear algebraic system is solved by preconditioned Krylov methods. The phase-field model, with a physically diffuse interface, affords the method several advantages in computing interfacial dynamics. One is the ease in simulating topological changes such as interfacial rupture and coalescence. Another is the capability of computing contact line motion without invoking ad hoc slip conditions. As validation of the 3D numerical scheme, we have computed drop deformation in an elongational flow, relaxation of a deformed drop to the spherical shape, and drop spreading on a partially wetting substrate. The results are compared with numerical and experimental results in the literature as well as our own axisymmetric computations where appropriate. Excellent agreement is achieved provided that the 3D interface is adequately resolved by using a sufficiently thin diffuse interface and refined grid. Since our model involves several coupled partial differential equations and we use a fully implicit scheme, the matrix inversion requires a large memory. This puts a limit on the scale of problems that can be simulated in 3D, especially for viscoelastic fluids.

Using a signed distance function for the simulation of metal forming processes: Formulation of the contact condition and mesh adaptation. From a Lagrangian approach to an Eulerian approach

AbstractThis paper proposes to use the metric properties of the distance function between two bodies in contact (or gap function) in simulations involving contact problems. First, the normal vectors, which are involved in the formulation of the contact condition, are defined through the gradient of this distance function. This definition avoids to deal with the numerical penetration parameter, which is generally introduced otherwise. Furthermore, it allows the contact problem to be extended in a simple way to an Eulerian formulation. Second, this paper investigates two mesh adaptation strategies based on the properties of the distance function. The first strategy consists in building a size map according to the values of this function, in order to refine locally the mesh, and consequently to improve the description of the contact surface. The second strategy consists in adapting locally the mesh to the geometry of the contact surface. This anisotropic adaptation is performed by constructing a metric map that allows the mesh size to be imposed in the direction of the distance function gradient. A lot of elements are saved when compared with the isotropic case. Throughout this paper, many numerical simulations are presented in the context of the forging process: the deformable material is pressed between two rigid tools. Furthermore, the algorithm used to calculate the signed distance to a surface mesh is detailed in appendix of this paper. Copyright © 2008 John Wiley & Sons, Ltd.

Goal‐oriented mesh adaptation for FE‐vibration analyses of shell‐like structures

AbstractThe well–known semidiscrete approach in structural dynamics consists of two steps. First the discretization of the spatial domain using finite elements is performed. In the second step the numerical integration of the resulting system of ordinary differential equations is executed. The numerical schemes in both discretization steps contribute to the total discretization error. In this contribution the focus is on the spatial discretization error and on efficient strategies for error assessment and error reduction by goal–oriented mesh adaptation schemes. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Adaptive and anisotropic mesh strategy for thin shell problems. Case of inhibited parabolic shells

The aim of this paper is to show the reliability of an adaptive and anisotropic mesh procedure for thin shell problems. We consider singular perturbation problems only for parabolic shells whose behavior is described by the Koiter model. The corresponding system of equations, which depends on the relative thickness ε of the shell, is elliptic except at the limit for ε = 0 where it is parabolic. In a first part of this paper, we study theoretically the phenomena of internal layers appearing during the singular perturbation process, when the loading is somewhat singular. These layers have very different structures either they are along or across the asymptotic lines of the middle surface of the shell. In a second part, numerical computations are performed using a finite element software coupled with an adaptive anisotropic mesh generator. This technique enables to approach accurately the singularities and the layers predicted by the theory especially for very small values of the thickness. The efficiency of such a procedure in comparison with uniform meshes is put in a prominent position.

An adaptive mesh strategy for high compressible flows based on nodal re-allocation

An adaptive mesh strategy based on nodal re-allocation is presented in this work. This technique is applied to problems involving compressible flows with strong shocks waves, improving the accuracy and efficiency of the numerical solution. The initial mesh is continuously adapted during the solution process keeping, as much as possible, mesh smoothness and local orthogonality using an unconstrained nonlinear optimization method. The adaptive procedure, which is coupled to an edge-based error estimate aiming to equidistribute the error over the cell edges is the main contribution of this work. The flow is simulated using the Finite Element Method (FEM) with an explicit one-step Taylor-Galerkin scheme, in which an Arbitrary Lagrangean-Eulerian (ALE) description is employed to take into account mesh movement. Finally, to demonstrate the capabilities of the adaptive process, several examples of compressible inviscid flows are presented.

Levelsets and anisotropic mesh adaptation

In this paper, we focus on the problem of adapting dynamic triangulations during numerical simulations to reduce the approximation errors. Dynamically evolving interfaces arise in many applications, such as free surfaces in multiphase flows and moving surfaces in fluid-structure interactions. In such simulations, it is often required to preserve a high quality interface discretization thus posing significant challenges in adapting the triangulation in the vicinity of the interface, especially if its geometry or its topology changes dramatically during the simulation. Our approach combines an efficient levelset formulation to represent the interface in the flow equations with an anisotropic mesh adaptation scheme based on a Riemannian metric tensor to prescribe size, shape and orientation of the elements. Experimental results are provided to emphasize the effectiveness of this technique for dynamically evolving interfaces in flow simulations.

Upper and lower bounds in limit analysis: Adaptive meshing strategies and discontinuous loading

AbstractUpper and lower bounds of the collapse load factor are here obtained as the optimum values of two discrete constrained optimization problems. The membership constraints for Von Mises and Mohr–Coulomb plasticity criteria are written as a set of quadratic constraints, which permits one to solve the optimization problem using specific algorithms for Second‐Order Conic Program (SOCP). From the stress field at the lower bound and the velocities at the upper bound, we construct a novel error estimate based on elemental and edge contributions to the bound gap. These contributions are employed in an adaptive remeshing strategy that is able to reproduce fan‐type mesh patterns around points with discontinuous surface loading. The solution of this type of problems is analysed in detail, and from this study some additional meshing strategies are also described. We particularise the resulting formulation and strategies to two‐dimensional problems in plane strain and we demonstrate the effectiveness of the method with a set of numerical examples extracted from the literature. Copyright © 2008 John Wiley & Sons, Ltd.

Two-dimensional finite element method for stress intensity factor using adaptive mesh strategy

Finite element analysis (FEA) combined with the concepts of Linear Elastic fracture mechanics (LEFM) provides a practical and convenient means to study the fracture and crack growth of materials. A numerical analysis (FEM) of cracks was developed to derive the SIF for two different geometries, i.e., a rectangular plate with half circle-hole and central edge crack plate in tension loading conditions. The onset criterion of crack propagation is based on the stress intensity factor, which is the most important parameter that must be accurately estimated and facilitated by the singular element. Displacement extrapolation technique (DET) is employed, to obtain the stress intensity factors (SIFs) at crack tip. The fracture is modeled by the splitting node approach and the trajectory follows the successive linear extensions of each crack increment. These comprehensive tests are evaluated and compared with other relevant numerical and analytical results obtained by other researchers.

Truncated Total Least Squares: A New Regularization Method for the Solution of ECG Inverse Problems

The reconstruction of epicardial potentials (EPs) from body surface potentials (BSPs) can be characterized as an ill-posed inverse problem which generally requires a regularized numerical solution. Two kinds of errors/noise: geometric errors and measurement errors exist in the ECG inverse problem and make the solution of such problem more difficulty. In particular, geometric errors will directly affect the calculation of transfer matrix A in the linear system equation AX = B. In this paper, we have applied the truncated total least squares (TTLS) method to reconstruct EPs from BSPs. This method accounts for the noise/errors on both sides of the system equation and treats geometric errors in a new fashion. The algorithm is tested using a realistically shaped heart-lung-torso model with inhomogeneous conductivities. The h-adaptive boundary element method [h-BEM, a BEM mesh adaptation scheme which starts from preset meshes and then refines (adds/removes) grid with fixed order of interpolation function and prescribed numerical accuracy] is used for the forward modeling and the TTLS is applied for inverse solutions and its performance is also compared with conventional regularization approaches such as Tikhonov and truncated single value decomposition (TSVD) with zeroth-, first-, and second-order. The simulation results demonstrate that TTLS can obtain similar results in the situation of measurement noise only but performs better than Tikhonov and TSVD methods where geometric errors are involved, and that the zeroth-order regularization is the optimal choice for the ECG inverse problem. This investigation suggests that TTLS is able to robustly reconstruct EPs from BSPs and is a promising alternative method for the solution of ECG inverse problems.