Adaptive Finite Element Procedure Research Articles

Error Analysis of a Decoupled Finite Element Method for Quad-Curl Problems

Finite element approximation to a decoupled formulation for the quad--curl problem is studied in this paper. The difficulty of constructing elements with certain conformity to the quad--curl problems has been greatly reduced. For convex domains, where the regularity assumption holds for Stokes equation, the approximation to the curl of the true solution has quadratic order of convergence and first order for the energy norm. If the solution shows singularity, an a posterior error estimator is developed and a separate marking adaptive finite element procedure is proposed, together with its convergence proved. Both the a priori and a posteriori error analysis are supported by the numerical examples.

A general adaptive finite element eigen-algorithm stemming from Wittrick-Williams algorithm

Stemming from Wittrick-Williams algorithm for transcendental eigenproblems, a general adaptive finite element (FE) procedure is proposed for general linear eigenproblems. The pivot of the proposed procedure is the counting method based on Sturm sequences, a simplified version of the Wittrick-Williams algorithm for linear matrix eigenproblems, based on which a general eigen-algorithm is developed by integrating a number of effective approaches such as a two-phased divide-and-conquer scheme, a guided-and-guarded technique for eigenvalue seeking, the inverse and subspace iterations for mode finding, etc. The adopted error estimator in the adaptive FE procedure is the recently developed one which is based on the element energy projection (EEP) technique for pointwise error estimation and hence is able to control errors by maximum norm with local mesh refinement. Representative numerical examples of plates and shells are given to show the validity, reliability and efficiency of the proposed procedure.

Dynamic adaptive finite element analysis of acoustic wave propagation due to underwater explosion for fluid-structure interaction problems

In this paper, an investigation into the propagation of far field explosion waves in water and their effects on nearby structures are carried out. For the far field structure, the motion of the fluid surrounding the structure may be assumed small, allowing linearization of the governing fluid equations. A complete analysis of the problem must involve simultaneous solution of the dynamic response of the structure and the propagation of explosion wave in the surrounding fluid. In this study, a dynamic adaptive finite element procedure is proposed. Its application to the solution of a 2D fluid-structure interaction is investigated in the time domain. The research includes: a) calculation of the far-field scatter wave due to underwater explosion including solution of the time-depended acoustic wave equation, b) fluid-structure interaction analysis using coupled Euler-Lagrangian approach, and c) adaptive finite element procedures employing error estimates, and re-meshing. The temporal mesh adaptation is achieved by local regeneration of the grid using a time-dependent error indicator based on curvature of pressure function. As a result, the overall response is better predicted by a moving mesh than an equivalent uniform mesh. In addition, the cost of computation for large problems is reduced while the accuracy is improved.

2.5D induced polarization forward modeling using the adaptive finite-element method

The conventional finite-element (FE) method often uses a structured mesh, which is designed according to the user’s experience, and it is not sufficiently accurate and flexible to accommodate complex structures such as dipping interfaces and rough topography. We present an adaptive FE method for 2.5D forward modeling of induced polarization (IP). In the presented method, an unstructured triangulation mesh that allows for local mesh refinement and flexible description of arbitrary model geometries is used. Furthermore, the mesh refinement process is guided by dual error estimate weighting to bias the refinement towards elements that affect the solution at the receiver locations. After the final mesh is generated, the Jacobian matrix is used to obtain the IP response on 2D structure models. We validate the adaptive FE algorithm using a vertical contact model. The validation shows that the elements near the receivers are highly refined and the average relative error of the potentials converges to 0.4 % and 1.2 % for the IP response. This suggests that the numerical solution of the adaptive FE algorithm converges to an accurate solution with the refined mesh. Finally, the accuracy and flexibility of the adaptive FE procedure are also validated using more complex models.

An adaptive finite element procedure for fully-coupled point contact elastohydrodynamic lubrication problems

This paper presents an automatic locally adaptive finite element solver for the fully-coupled EHL point contact problems. The proposed algorithm uses a posteriori error estimation in the stress in order to control adaptivity in both the elasticity and lubrication domains. The implementation is based on the fact that the solution of the linear elasticity equation exhibits large variations close to the fluid domain on which the Reynolds equation is solved. Thus the local refinement in such region not only improves the accuracy of the elastic deformation solution significantly but also yield an improved accuracy in the pressure profile due to increase in the spatial resolution of fluid domain. Thus, the improved traction boundary conditions lead to even better approximation of the elastic deformation. Hence, a simple and an effective way to develop an adaptive procedure for the fully-coupled EHL problem is to apply the local refinement to the linear elasticity mesh. The proposed algorithm also seeks to improve the quality of refined meshes to ensure the best overall accuracy. It is shown that the adaptive procedure effectively refines the elements in the region(s) showing the largest local error in their solution, and reduces the overall error with optimal computational cost for a variety of EHL cases. Specifically, the computational cost of proposed adaptive algorithm is shown to be linear with respect to problem size as the number of refinement levels grows.

An Adaptive FEM for Buckling Analysis of Nonuniform Bernoulli-Euler Members via the Element Energy Projection Technique

This paper presents a new adaptive finite element (FE) procedure for accurate, efficient, and reliable computation of the critical buckling loads and the associated modes of nonuniform Bernoulli-Euler members. After the conventional FE solution on a given mesh has been obtained, a novel conceptual and practical strategy is introduced, in which the FE solution of the eigenproblem is equivalently viewed as the FE solution of an associated linear problem. This strategy allows the recently developed element energy projection (EEP) technique to be readily used to calculate super-convergent FE solutions for buckling modes, which are subsequently used to estimate the FE solution errors and to further guide mesh refinements, until the accuracy of the FE solution matches the user-preset error tolerance. Numerical examples are given to show the effectiveness, efficiency, accuracy, and reliability of the proposed method, which also paves the way for development of an adaptive FE solver for more general and complicated eigenproblems.

Refined h-adaptive finite element procedure for large deformation geotechnical problems

Adaptive finite element procedures automatically refine, coarsen, or relocate elements in a finite element mesh to obtain a solution with a specified accuracy. Although a significant amount of research has been devoted to adaptive finite element analysis, this method has not been widely applied to nonlinear geotechnical problems due to their complexity. In this paper, the h-adaptive finite element technique is employed to solve some complex geotechnical problems involving material nonlinearity and large deformations. The key components of h-adaptivity including robust mesh generation algorithms, error estimators and remapping procedures are discussed. This paper includes a brief literature review as well as formulation and implementation details of the h-adaptive technique. Finally, the method is used to solve some classical geotechnical problems and results are provided to illustrate the performance of the method.

Modeling of cohesive crack growth using an adaptive mesh refinement via the modified-SPR technique

In this paper, an adaptive finite element procedure is presented in modeling of mixed-mode cohesive crack propagation via the modified superconvergent path recovery technique. The adaptive mesh refinement is performed based on the Zienkiewicz–Zhu error estimator. The weighted-SPR recovery technique is employed to improve the accuracy of error estimation. The Espinosa–Zavattieri bilinear cohesive zone model is applied to implement the traction-separation law. It is worth mentioning that no previous information is necessary for the path of crack growth and no region of the domain is necessary to be filled by the cohesive elements. The maximum principal stress criterion is employed for predicting the direction of extension of the cohesive crack in order to implement the cohesive elements. Several numerical examples are analyzed numerically to demonstrate the capability and efficiency of proposed computational algorithm.

Adaptive finite element modelling of two-dimensional magnetotelluric fields in general anisotropic media

An adaptive unstructured mesh finite element (FE) procedure is presented for improving the quality of numerical solutions to the magnetotelluric forward problem in a general 2-D anisotropic conductivity structure. We implement a self-adaptive, goal-oriented grid refinement algorithm in which a finite element analysis is performed on a sequence of refined meshes. The mesh refinement process is guided by a dual error estimate weighting to bias refinement towards elements that affect the solution at the EM receiver locations. We validate the finite element code against a layered 1-D model with a sea water layer. Further, we compare the FE results with those obtained by a finite-difference (FD) scheme for both a block seamountain and a sea bottom hill model. Both FE and FD schemes show very good agreement for the block seamountain model. For the sea bottom hill model, however, only on the flat seafloor segments both the FE and FD solutions fit very well, but on the seafloor slope, FD results are oscillating due to a simplistic staircase approximation of the bathymetric undulations. The FD scheme for 2-D anisotropic conductors, developed primarily for the modelling of magnetotelluric data on a flat Earth surface, is thus not an adequate tool for dealing with structures with sloping bathymetry and topography, whereas the FE method with adapting mesh can easily handle such structures at almost any level of complexity.

Adaptive finite element methods in geodynamics

PurposeThe purpose of this paper is to present an adaptive finite element procedure that improves the quality of convection dominated mid‐ocean ridge (MOR) and subduction zone (SZ) simulations in geodynamics.Design/methodology/approachThe method adapts the mesh automatically around regions of high‐solution gradient, yielding enhanced resolution of the associated flow features. The approach utilizes an automatic, unstructured mesh generator and a finite element flow solver. Mesh adaptation is accomplished through mesh regeneration, employing information provided by an interpolation‐based local error indicator, obtained from the computed solution on an existing mesh.FindingsThe proposed methodology works remarkably well at improving solution accuracy for both MOR and SZ simulations. Furthermore, the method is computationally highly efficient.Originality/valueTo date, successful goal‐orientated/error‐guided grid adaptation techniques have, to the knowledge, not been utilized within the field of geodynamics. This paper presents the first true geodynamical application of such methods.

Investigations into the applicability of adaptive finite element methods to two‐dimensional infinite Prandtl number thermal and thermochemical convection

An adaptive finite element procedure is presented for improving the quality of solutions to convection‐dominated problems in geodynamics. The method adapts the mesh automatically around regions of high solution gradient, yielding enhanced resolution of the associated flow features. The approach requires the coupling of an automatic mesh generator, a finite element flow solver, and an error estimator. In this study, the procedure is implemented in conjunction with the well‐known geodynamical finite element code ConMan. An unstructured quadrilateral mesh generator is utilized, with mesh adaptation accomplished through regeneration. This regeneration employs information provided by an interpolation‐based local error estimator, obtained from the computed solution on an existing mesh. The technique is validated by solving thermal and thermochemical problems with well‐established benchmark solutions. In a purely thermal context, results illustrate that the method is highly successful, improving solution accuracy while increasing computational efficiency. For thermochemical simulations the same conclusions can be drawn. However, results also demonstrate that the grid‐based methods employed for simulating the compositional field are not competitive with the other methods (tracer particle and marker chain) currently employed in this field, even at the higher spatial resolutions allowed by the adaptive grid strategies.

Simulation of corrosion fatigue crack growth under mixed-mode loading

The kinking of a corrosion crack due to mixed-mode fatigue loading is studied using an adaptive finite element procedure. The rate of material dissolution is assumed to be proportional to the stretching of the corroding surface. The dissolution of material is governed by a corrosion law, where no criterion is needed for neither crack growth nor growth direction. The problem is treated as a general moving boundary problem. The kink angles are found to be in very good agreement with results for sharp cracks using criteria reported in the literature.

An adaptive finite element procedure for crack propagation analysis

This paper presents the adaptive mesh finite element estimation method for analyzing 2D linear elastic fracture problems. The mesh is generated by the advancing front method and the norm stress error is taken as a posteriori error estimator for the h-type adaptive refinement. The stress intensity factors are estimated by a displacement extrapolation technique. The near crack tip displacements used are obtained from specific nodes of natural six-noded quarter-point elements which are generated around the crack tip defined by the user. The crack growth and its direction are determined by the calculated stress intensity factors. The maximum circumference theory is used for the latter. In evaluating the accuracy of the estimated stress intensity factors, four cases are tested consisting of compact tension specimen, three-point bending specimen, central cracked plate and double edge notched plate. These were carried out and compared to the results from other studies. The crack trajectories of these specimen tests are also illustrated.

A contribution to error estimation and mapping algorithms for a hierarchical adaptive FE-strategy in finite elastoplasticity

The aim of this contribution is the presentation of an adaptive finite element procedure for the solution of geometrically and physically non-linear problems in structural mechanics. Within this context, the attention is mainly directed on the error estimation and hierarchical strategies for mesh refinement and coarsening in the case of finite elasto-plastic deformations. An important but sensitive aspect of adaptation approaches of the space discretization is the calculation of mechanical field variables for the modified mesh. Procedures of mesh refinement and coarsening imply the determination of strains, stresses and internal variables at the nodes and the Gauss points of new elements based on the transfer of the required data from the former mesh. In order to improve the efficiency as well as the convergence behaviour of the adaptive FE process an approach of data transfer primarily related to nodal values is presented. It is characterized by solving the initial value problem not only at the Gauss points but additionally at the nodes of the elements.

Adaptive superposition of finite element meshes in elastodynamic problems

An adaptive finite element procedure is developed for modelling transient phenomena in elastic solids, including both wave propagation and structural dynamics. Although both temporal and spatial adaptivity are addressed, the novel feature of the formulation is the use of mesh superposition to produce spatial refinement (referred to as s-adaptivity) in transient problems. Spatial error estimation is based on superconvergent patch recovery of higher-order accurate stresses and is used to guide mesh adaptivity, while the temporal error estimation is based on the assumption of linearly varying third-order time derivatives of the displacement field and is used to adjust the time step size for the HHT-α variant of the Newmark direct numerical integration method. Spatial adaptivity of the mesh is performed using a hierarchical h-refinement scheme that is efficiently implemented using a structured version of finite element mesh superposition. This particular spatial adaptivity scheme is extremely fast and consequently makes it feasible to repeatedly update both the mesh and the time increment as required in an adaptive transient analysis. This work represents the initial effort in applying this type of spatial adaptivity to transient problems. Three example problems are given to demonstrate the performance characteristics of the s-adaptive procedure. Copyright © 2005 John Wiley & Sons, Ltd.

Adaptive finite element procedures for elastoplastic problems at finite strains

AbstractA major difficulty in the context of adaptive analysis of geometrically nonlinear problems is to provide a robust remeshing procedure that accounts both for the error caused by the spatial discretization and for the error due to the time discretization. For stability problems, such as strain localization and necking, it is essential to provide a step–size control in order to get a robust algorithm for the solution of the boundary value problem. For this purpose we developed an easy to implement step–size control algorithm. In addition we will consider possible a posteriori error indicators for the spatial error distribution of elastoplastic problems at finite strains. This indicator is adopted for a density–function–based adaptive remeshing procedure. Both error indicators are combined for the adaptive analysis in time and space. The performance of the proposed method is documented by means of representative numerical examples.

Adaptive Finite–Element Procedures for Homogenization Analysis of Heterogeneous Micro–Structures

Adaptive Finite–Element Procedures for Homogenization Analysis of Heterogeneous Micro–Structures

A plasticity based model and an adaptive algorithm for finite element analysis of reinforced concrete panels

AbstractThis paper deals with an adaptive finite element procedure for the analysis of plain and reinforced concrete panels in a state of plane stress. Therefore, we will present a plasticity based model for plain concrete which captures the two failure modes of concrete within one formulation. In spite of a simple formulation the model is capable to describe the different mechanisms for tensile failure as well as for compression fracture. To restrict the time discretization error and the spatial discretization error to certain tolerances, the constitutive model is embedded in an adaptive algorithm which controls the size of the incremental load steps and leads to a hierarchical mesh refinement if necessary. The application of the model will be shown by various numerical examples. Copyright © 2002 John Wiley & Sons, Ltd.

An adaptive finite element procedure for solidification problems

An adaptive mesh regeneration procedure for transient solidification problems has been presented in this paper. The method is based on the local minimum and maximum values of second derivatives of velocity and temperature fields. A numerical example of alloy solidification in a square enclosure is presented for demonstration. Extension to more complicated and odd-shaped problems is straight forward.

A NUMERICAL SIMULATION MODEL FOR THE GROWTH OF GaxIn1-xSb BY THE TRAVELLING HEATER METHOD

A numerical simulation model for the growth of Ga-xIn1-xSb by the traveling heater method (THM) is presented. The cell configuration and the furnace thermal profile are adopted from the experimental setup of Amistar Research inc. The field equations are solved numerically by an adaptive finite element procedure as the interfaces between the solid and liquid phases change in time. Numerical results show that the furnace thermal profile, the thermal as well as solutal convection in the liquid solution have significant effects on the growth process. Results are only presented for small growth times. However, it is asserted that the present simulation model could be used to simulate the entire THM growth process of ternary alloys if the required computational power is available.