Adaptive Finite Element Mesh Research Articles

Efficient All-Electron Time-Dependent Density Functional Theory Calculations Using an Enriched Finite Element Basis.

We present an efficient and systematically convergent approach to all-electron real-time time-dependent density functional theory (TDDFT) calculations using a mixed basis, termed as enriched finite element (EFE) basis. The EFE basis augments the classical finite element basis (CFE) with a compactly supported numerical atom-centered basis, obtained from atomic ground-state DFT calculations. Particularly, we orthogonalize the enrichment functions with respect to the classical finite element basis to ensure good conditioning of the resultant basis. We employ the second-order Magnus propagator in conjunction with an adaptive Krylov subspace method for efficient time evolution of the Kohn-Sham orbitals. We rely on a priori error estimates to guide our choice of an adaptive finite element mesh as well as the time step to be used in the TDDFT calculations. We observe close to optimal rates of convergence of the dipole moment with respect to spatial and temporal discretizations. Notably, we attain a 50-100 times speedup for the EFE basis over the CFE basis. We also demonstrate the efficacy of the EFE basis for both linear and nonlinear responses by studying the absorption spectra in sodium clusters, the linear to nonlinear response transition in the green fluorescence protein chromophore, and the higher harmonic generation in the magnesium dimer. Lastly, we attain good parallel scalability of our numerical implementation of the EFE basis for up to ∼1000 processors, using a benchmark system of a 50-atom sodium nanocluster.

Adaptive Finite Element Modeling of Linear Elastic Fatigue Crack Growth.

This paper proposed an efficient two-dimensional fatigue crack growth simulation program for linear elastic materials using an incremental crack growth procedure. The Visual Fortran programming language was used to develop the finite element code. The adaptive finite element mesh was generated using the advancing front method. Stress analysis for each increment was carried out using the adaptive mesh finite element technique. The equivalent stress intensity factor is the most essential parameter that should be accurately estimated for the mixed-mode loading condition which was used as the onset criterion for the crack growth. The node splitting and relaxation method advances the crack once the failure mechanism and crack direction have been determined. The displacement extrapolation technique (DET) was used to calculate stress intensity factors (SIFs) at each crack extension increment. Then, these SIFs were analyzed using the maximum circumferential stress theory (MCST) to predict the crack propagation trajectory and the fatigue life cycles using the Paris' law model. Finally, the performance and capability of the developed program are shown in the application examples.

A Gradient-Free Topology Optimization Strategy for Continuum Structures with Design-Dependent Boundary Loads

In this paper, the topology optimization of continuum structures with design-dependent loads is studied with a gradient-free topology optimization method in combination with adaptive body-fitted finite element mesh. The material-field series-expansion (MFSE) model represents the structural topology using a bounded material field with specified spatial correlation and provides a crisp structural boundary description. This feature makes it convenient to identify the loading surface for the application of the design-dependent boundary loads and to generate a body-fitted mesh for structural analysis. Using the dimension reduction technique, the number of design variables is significantly decreased, which enables the use of an efficient Kriging-based algorithm to solve the topology optimization problem. The effectiveness of the proposed method is demonstrated using several numerical examples, among which a design problem with geometry and contact nonlinearity is included.

Fast Adaptive Augmented Lagrangian Digital Image Correlation

Digital image correlation (DIC) is a powerful experimental technique for measuring full-field displacement and strain. The basic idea of the method is to compare images of an object decorated with a speckle pattern before and after deformation in order to compute the displacement and strain fields. Local Subset DIC and finite element-based Global DIC are two widely used image matching methods; however there are some drawbacks to these methods. In Local Subset DIC, the computed displacement field may not satisfy compatibility, and the deformation gradient may be noisy, especially when the subset size is small. Global DIC incorporates displacement compatibility, but can be computationally expensive. In this thesis, we propose a new method, the augmented-Lagrangian digital image correlation (ALDIC), that combines the advantages of both the local (fast and in parallel) and global (compatible) methods. We demonstrate that ALDIC has higher accuracy and behaves more robustly compared to both Local Subset DIC and Global DIC. DIC requires a large number of high resolution images, which imposes significant needs on data storage and transmission. We combined DIC algorithms with image compression techniques and show that it is possible to obtain accurate displace- ment and strain fields with only 5 % of the original image size. We studied two compression techniques – discrete cosine transform (DCT) and wavelet transform, and three DIC algorithms – Local Subset DIC, Global DIC and our newly proposed augmented Lagrangian DIC (ALDIC). We found the Local Subset DIC leads to the largest errors and ALDIC to the smallest when compressed images are used. We also found wavelet-based image compression introduces less error compared to DCT image compression. To further speed up and improve the accuracy of DIC algorithms, especially in the study of complex heterogeneous strain fields at various length scales, we apply an adaptive finite element mesh to DIC methods. We develop a new h-adaptive technique and apply it to ALDIC. We show that this adaptive mesh ALDIC algorithm significantly decreases computation time with no loss (and some gain) in accuracy.

Gradient flow finite element discretizations with energy-based adaptivity for the Gross-Pitaevskii equation

We present an effective adaptive procedure for the numerical approximation of the steady-state Gross–Pitaevskii equation. Our approach is solely based on energy minimization, and consists of a combination of a novel adaptive finite element mesh refinement technique, which does not rely on any a posteriori error estimates, and a recently proposed new gradient flow. Numerical tests show that this strategy is able to provide highly accurate results, with optimal convergence rates with respect to the number of degrees of freedom.

Research on hilly mountain tractor based on adaptive

Finite element method is an effective numerical analysis method developed to adapt to the use of computer [1]. Its role in engineering practice has been extended from analysis and verification to optimization design, and it has become a computer aided by computer aided design (CAD) Engineering (CAE) an important part. With the increasing complexity of engineering practice problems, the application of finite element method and error analysis method in one of the adaptive finite element method is becoming more and more popular. The adaptive finite element mesh which reflects the physical and geometric characteristics of the structure is the necessary part of applying the adaptive finite element method. With the trend of the development of tractors in the direction of intelligent, lightweight, the application of electronic technology and micro-control technology to tractors is an inevitable way. In this paper, based on the traditional tractor hydraulic suspension system, a tractor system with force adjustment function is designed. Compared with the traditional tractor hydraulic suspension system, the system uses the electronic control technology to realize the deep displacement adjustment and force regulation, control the chip acquisition force sensor and the position sensor real-time feedback signal to raise or lower the hanging rod. Introduction.

The role of sedimentation and natural compaction in a prograding delta: insights from the mega Mekong delta, Vietnam

The Vietnamese Mekong Delta was formed by rapid transgression during the second half of the Holocene by deposition of mainly unconsolidated, fine-grained (clayey) sediments undergoing high compaction rates. The natural subsidence can seriously impact the already vulnerable delta plain as its low elevation exposes the delta to global sea level rise, flooding, salinization. Human activities such as groundwater pumping, infrastructural loading, sand mining and dam construction have exacerbated the effects of natural consolidation. Here we present a novel modeling study that has allowed to reproduce the formation and evolution of the Mekong delta over the past 4000 years. Using an adaptive finite-element mesh, the model properly simulates accretion and natural consolidation characterizing the delta evolution. Large soil grain motion and the delayed dissipation of pore-water overpressure are accounted for. We find that natural compaction of Holocene deposits following delta evolution exceeds predicted values of absolute sea level rise. The unprecedented high rates (up to ~20 mm/yr) threaten the lower delta plain with permanent inundation and inevitably reduce the designed service life of flood defense structures along the coast. Total subsidence and sediment delivery to the delta plain will determine its future elevation and vulnerability to relative sea level rise.

An efficient 2D inversion scheme for airborne frequency-domain data

In many cases, inversion in 2D gives a better description of the subsurface compared with 1D inversion, but, computationally, 2D inversion is expensive, and it can be hard to use for large-scale surveys. We have developed an efficient hybrid 2D airborne frequency-domain electromagnetic inversion algorithm. Our hybrid scheme combines 1D and 2D inversions in a three-stage process, in which each step is progressively more accurate and computationally more expensive than the previous one. This results in an approximately [Formula: see text] speedup compared with full 2D inversions, and with only minor changes to the inversion results. Our inversion structure is based on a regular grid, in which each sounding is discretized individually. The 1D modeling code uses layered models with derivatives derived through the finite-difference method, whereas our 2D modeling code uses an adaptive finite-element mesh, and it uses the adjoint-state method to calculate the derivatives. By incorporating the inversion grid structure into the 2D finite-element mesh, interpolation between the different meshes becomes trivial. Large surveys are handled by using local meshing to split large surveys into small sections, which retains the 2D information. The algorithm is heavily optimized and parallelized over the frequencies and sections, with good scalability even on nonuniform memory architecture systems, on which it is generally hard to achieve a satisfactory scaling. The algorithm has been tested successfully with various synthetic studies as well as field examples, of which results from two synthetic studies and a field example are shown.

Numerical simulations of Holocene salt-marsh dynamics under the hypothesis of large soil deformations

Abstract Salt marshes are vulnerable environments hosting complex interactions between physical and biological processes. The prediction of the elevation dynamics of a salt-marsh platform is crucial to forecast its future behavior under potential changing scenarios. An original finite-element (FE) numerical model accounting for the long-term marsh accretion and compaction linked to relative sea level rise is proposed. The accretion term considers the material sedimentation over the marsh surface, whereas the compaction reflects the progressive consolidation of the porous medium under the increasing load of the overlying younger deposits. The modeling approach is based on a 2D groundwater flow simulator coupled to a 1D vertical geomechanical module, where the soil properties may vary with the effective intergranular stress. The model takes also into account the geometric non-linearity arising from the consideration of large solid grain movements by using a Lagrangian approach with an adaptive FE mesh. The numerical experiments show the potentiality of the proposed 2D model, which consistently integrates in modeling framework the behavior of spatially distributed model parameters. High sedimentation rates and low permeabilities largely impact on the mechanism of soil compaction following the overpressure dissipation.

An adaptive Fixed-Mesh ALE method for free surface flows

In this work we present a Fixed-Mesh ALE method for the numerical simulation of free surface flows capable of using an adaptive finite element mesh covering a background domain. This mesh is successively refined and unrefined at each time step in order to focus the computational effort on the spatial regions where it is required. Some of the main ingredients of the formulation are the use of an Arbitrary-Lagrangian–Eulerian formulation for computing temporal derivatives, the use of stabilization terms for stabilizing convection, stabilizing the lack of compatibility between velocity and pressure interpolation spaces, and stabilizing the ill-conditioning introduced by the cuts on the background finite element mesh, and the coupling of the algorithm with an adaptive mesh refinement procedure suitable for running on distributed memory environments. Algorithmic steps for the projection between meshes are presented together with the algebraic fractional step approach used for improving the condition number of the linear systems to be solved. The method is tested in several numerical examples. The expected convergence rates both in space and time are observed. Smooth solution fields for both velocity and pressure are obtained (as a result of the contribution of the stabilization terms). Finally, a good agreement between the numerical results and the reference experimental data is obtained.

Three-dimensional modeling of a thermal dendrite using the phase field method with automatic anisotropic and unstructured adaptive finite element meshing

Dendritic growth is computed with automatic adaptation of an anisotropic and unstructured finite element mesh. The energy conservation equation is formulated for solid and liquid phases considering an interface balance that includes the Gibbs-Thomson effect. An equation for a diffuse interface is also developed by considering a phase field function with constant negative value in the liquid and constant positive value in the solid. Unknowns are the phase field function and a dimensionless temperature, as proposed by [1]. Linear finite element interpolation is used for both variables, and discretization stabilization techniques ensure convergence towards a correct non-oscillating solution. In order to perform quantitative computations of dendritic growth on a large domain, two additional numerical ingredients are necessary: automatic anisotropic unstructured adaptive meshing [2,[3] and parallel implementations [4], both made available with the numerical platform used (CimLib) based on C++ developments. Mesh adaptation is found to greatly reduce the number of degrees of freedom. Results of phase field simulations for dendritic solidification of a pure material in two and three dimensions are shown and compared with reference work [1]. Discussion on algorithm details and the CPU time will be outlined.

An Adaptive Mesh Generation Scheme for the Finite Element Method

재난 상황에서 방재시스템의 구조물이 제대로 역할을 수행하려면 구조의 자동화된 상태 파악 기능이 필수적이다. 이 기능은 실시간 연산면에서 매우 효율적이어야 한다. 유한요소법은 구조해석법으로 가장 많이 사용되는 방법으로 자리 잡고 있으며, 자동화된 구조물의 유한요소 해석에서의 요소망은 적응적 형성 기법으로 생산하여야 한다. 적응적 요소망 기능이 없으면, 같은 요소망으로 유한요소 해석을 수행하는데, 이 경우에는 불필요한 부분의 많은 요소에 의한 연산 비효율성이 있을 수 있고, 일반적으로 초기와 최종 요소망 모양만 검토하기 때문에 해석 중 생길 수 있는 과도하게 변형된 요소에 의한 오차 증가도 초래할 수 있다. 다소 복잡한 동적 및 비선형 해석에서 이러한 문제가 자주 발생한다. 본 연구는 시간영역 동적해석에서 전 단계 해석 결과를 사용하여 계산된 대표 변형률 값을 오차 평가에 사용하고, 요소 세분화는 절점 이동인 r-법과 요소 분할인 h-법의 효율적인 조합을 분산 매개변수(dispersion parameter)를 활용하여 계산하고, 모양계수(shape factor) 개념으로 과대하게 변형된 요소 모양을 방지하는 동적인 적응적 유한 요소망 형성 전략을 제시한다. 적용한 예제를 통하여 정확성과 연산 효율성을 검증하고 나아가 방법의 간단함이 지진 하중, 풍하중 및 비선형 해석이 필요한 방재 구조시스템의 복잡한 해석에도 효율적으로 사용될 수 있는 가능성을 보여 준다. Ideal structures in hazard mitigation systems must have automated state identification capabilities in order to function acceptably in case of a severe hazard. This capability must be efficient in terms of real time computations. The finite element method has become the most widely used method of structural analysis and in an automated finite element structural analysis, an adaptive mesh generation scheme has become an essential component. Without an adaptive scheme, the same mesh is used throughout the analysis and this causes computational inefficiency due to fine mesh where they are not needed and inaccuracy due to overly distorted elements formed during the analysis that may proceed undetected as only the initial and the final element shapes are generally checked in practice. This is especially true for dynamic and nonlinear problems in structural analyses. Thus, the finite element mesh for these types of analyses must be dynamically adaptive and computationally efficient. In this paper, an efficient adaptive finite element mesh generation scheme for dynamic analyses of planar problems is described. Representative strain values are used for error estimates and refinements of meshes use combinations of the h-method (node movement) and the r-method (element division) based on a dispersion parameter. A coefficient that depends on the shape of element is used to correct overly distorted elements. A case study shows the validity and computational efficiency of the scheme. The study also demonstrates the potential applicability of the scheme's effective use in complex structural problems such as those under severe environmental hazards such as seismic loads, erratic wind loads, and general nonlinear problems.

3D Coupled Cellular Automaton (CA)–Finite Element (FE) Modeling for Solidification Grain Structures in Gas Tungsten Arc Welding (GTAW)

A coupled Cellular Automaton (CA) – Finite Element (FE) model is proposed to predict the grain structure formation during Gas Tungsten Arc Welding (GTAW). The FE model solves the heat flow problem based on an adaptive meshing. This is done on a first FE mesh. The CA model simulates the development of the envelope of the grains in the liquid. For that purpose, a second FE mesh, referred to as CA mesh, is used. Fields can be interpolated between the adaptive FE mesh and the CA mesh. A CA grid made of a regular lattice of cubic cells is defined and superimposed onto the CA mesh. A new dynamic strategy for the allocation/deallocation of the CA grid is proposed to reduce the computation and memory costs. This CAFE model is applied to partial melting of an initial grain structure and epitaxial growth in the undercooled zone of a liquid pool, thus simulating the formation of solidification structure during the GTAW process. Examples of single linear passes simulations for various processing conditions and a multiple pass simulation are presented.

Improved Numerical Determination of the Effective Stress within the Framework of Local Approach Cleavage Models

For the determination of the Weibull stress or analogous effective stress measures in the assessment of the cleavage failure probability based on finite element results, robust and numerically stable procedures are necessary. For the quantitative assessment of the numerical errors in the approximate integration procedures for the determination of the Weibull stress or corresponding stress measures, an appropriate error indicator in analogy to adaptive finite element meshing methods was derived. The mathematical considerations reveal that an evaluation of the cleavage probability by means on the classical Beremin (1983) model using the standard one-point Gauss integration scheme may result in distinct numerical errors. Therefore, an enhanced numerical integration formula was derived and implemented. This formula facilitates a numerical evaluation of the integrals with improved accuracy.

동적 문제에 효율적인 적응적 유한요소망

The finite element method has become the most widely used method of structural analysis and recently, the method has often been applied to complex dynamic and nonlinear structural analyses problems. Even for these complex problems, where the responses are hard to predict, finite element analyses yield reliable results if appropriate element types and meshes are used. However, the dynamic and nonlinear behaviors of a structure often include large deformations in various portions of the structure and if the same mesh is used throughout the analysis, some elements may deform to shapes beyond the reliable limits; thus dynamically adapting finite element meshes are needed in order for the finite element analyses to be accurate. In addition, to satisfy the users requirement of quick real run time of finite element programs, the algorithms must be computationally efficient. This paper presents an adaptive finite element mesh generation scheme for dynamic analyses of structures that may adapt at each time step. Representative strain values are used for error estimates and combinations of the h-method(node movement) and the r-method(element division) are used for mesh refinements. A coefficient that depends on the shape of an element is used to limit overly distorted elements. A simple frame example shows the accuracy and computational efficiency of the scheme. The aim of the study is to outline the adaptive scheme and to demonstrate the potential use in general finite element analyses of dynamic and nonlinear structural problems commonly encountered.

Adaptive Finite Element Mesh Generation for Dynamic Planar Problems

구조물의 전산 방재시스템의 한 부분인 설계-해석은 자동화가 필수적이다. 구조해석법으로는 유한요소법이 가장 많이 사용되는 방법으로 자리 잡고 있으며, 근래에는 다소 복잡한 동적인 문제와 비선형 문제에도 사용이 일반화 되고 있다. 구조물을 유한요소로 표현할 때 요소와 요소망이 적절히 선택되면 신뢰 있는 해석 결과를 출력한다. 그러나 구조물의 동적 또는 비선형 거동에는 예상하지 않은 부분에서 큰 변형이 일어날 수 있으며, 초기 요소망을 계속 사용하면 요소의 모양이 신뢰 있는 범위에서 상당히 벗어 날 수 있으므로 요소망 역시 동적으로 적응할 필요가 있고, 실시간으로 빠르게 진행되는 해석 과정을 고려하면, 동적인 적응적 유한 요소망 형성 전략은 연산면에서도 효율적이어야 한다. 본 연구는 시간영역 동적해석에서 전 단계 해석 결과를 사용하여 계산된 대표 변형률 값을 오차 평가에 사용하고, 요소 세분화는 절점 이동인 r-법과 요소 분할인 h-법의 조합으로 계산하며, 모양계수 개념으로 과대하게 변형된 요소 모양을 방지하는 동적인 적응적 유한 요소망 형성 전략을 제시한다. 적용한 캔틸레버 보의 예제를 통하여 정확성과 연산 효율성을 검증하였고 나아가 방법의 간단함이 지진 하중, 풍하중 등에 대한 해석이 필요한 방재 구조시스템의 복잡한 해석에도 효율적으로 사용될 수 있는 것을 보여 준다. In structural design and analysis used for hazard mitigation systems, automation is becoming an essential feature. For analysis, the finite element method has proven to be an effective approximate method if proper element types and meshes are chosen. Recently, the method has been successfully applied to solve complex dynamic and nonlinear problems; and with a properly chosen element type and mesh, reliable results have been obtained. However, in automation and in complex analyses of a structures, using the initial mesh throughout the analysis may involve some elements to go through strains beyond the elements' reliable limits. Thus, the finite element mesh for these types of analyses must be dynamically adaptive, and considering the rapid process of analysis in real time, the dynamically adaptive finite element mesh generating schemes must be computationally efficient. In this paper, a computationally efficient dynamically adaptive finite element mesh generation scheme for dynamic analyses of planar problems is described. The concept of representative strain value is used for error estimates and the refinements of meshes use combinations of the h-method (node movement) and the r-method (element division). A coefficient that depends on shape of elements is used to correct overly distorted elements. The validity of the scheme is shown by a deep cantilever beam example under a dynamic concentrated load. The example shows reasonable accuracy and efficient computing time. Furthermore, the study shows the potential for the scheme's effective use in complex structural problems such as those under severe environmental hazards such as seismic or erratic wind loads.

Refinement of flexible space–time finite element meshes and discontinuous Galerkin methods

In this paper we present an algorithm to refine space–time finite element meshes as needed for the numerical solution of parabolic initial boundary value problems. The approach is based on a decomposition of the space–time cylinder into finite elements, which also allows a rather general and flexible discretization in time. This also includes adaptive finite element meshes which move in time. For the handling of three-dimensional spatial domains, and therefore of a four-dimensional space–time cylinder, we describe a refinement strategy to decompose pentatopes into smaller ones. For the discretization of the initial boundary value problem we use an interior penalty Galerkin approach in space, and an upwind technique in time. A numerical example for the transient heat equation confirms the order of convergence as expected from the theory. First numerical results for the transient Navier–Stokes equations and for an adaptive mesh moving in time underline the applicability and flexibility of the presented approach.

Adaptive finite element mesh triangulation using self-organizing neural networks

The finite element method is a computationally intensive method. Effective use of the method requires setting up the computational framework in an appropriate manner, which typically requires expertise. The computational cost of generating the mesh may be much lower, comparable, or in some cases higher than the cost associated with the numeric solver of the partial differential equations, depending on the application and the specific numeric scheme at hand. The aim of this paper is to present a mesh generation approach using the application of self-organizing artificial neural networks through adaptive finite element computations. The problem domain is initially constructed using the self-organizing neural networks. This domain is used as the background mesh which forms the input for finite element analysis and from which adaptive parameters are calculated through adaptivity analysis. Subsequently, self-organizing neural network is used again to adjust the location of randomly selected mesh nodes as is the coordinates of all nodes within a certain neighborhood of the chosen node. The adjustment is a movement of the selected nodes toward a specific input point on the mesh. Thus, based on the results obtained from the adaptivity analysis, the movement of nodal points adjusts the element sizes in a way that the concentration of elements will occur in the regions of high stresses. The methods and experiments developed here are for two-dimensional triangular elements but seem naturally extendible to quadrilateral elements.

An Adaptive Finite Element Method for Minor Shape Modification in Optimization Algorithms

In this paper, we propose a method of adaptive finite element meshing, which is effective for minor shape modifications in optimization of electric machines. In the method, the mesh for the new shape in the optimization process is generated from the mesh of the previous shape with minimum modification. The locations of the nodes for the new shape are decided by solving the Laplace's equation, whose unknowns are displacements of the nodes. The adaptive finite element meshing is started from the modified mesh. The method can reduce the calculation time and improve the quality of the optimized results.

Adaptive finite element meshing at each iterative calculation for electromagnetic field analysis of rotating machines

AbstractIn this paper, we propose a novel algorithm of the nonlinear time‐stepping finite element method for analyses of rotating machines. In the method, the adaptive finite element meshing is applied at each nonlinear and time‐stepping iterative calculation with the interpolate calculations of the potentials from the previous step. The primary finite element mesh at each time step is obtained from the final mesh of the previous time step with minor modification due to the movement of the rotor. The proposed method is applied to the characteristics calculations of a permanent magnet motor and an induction motor to verify the method's validity and its usefulness. It is clarified that the proposed method gives almost identical results to the conventional adaptive meshing method whereas the calculation time is reduced to less than 1/10 in most cases. As a result, the proposed method can reduce not only the time of making mesh manually but also the calculation time compared with the usual finite element method without adaptive meshing. © 2008 Wiley Periodicals, Inc. Electr Eng Jpn, 164(3): 78–91, 2008; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/eej.20308