Octasection‐based refinement of finite element approximations of tetrahedral meshes that guarantees shape quality

During the last 2 years, a multidomain formalism for structural dynamics based on a multi‐time‐step algorithm and local linear modal reduction was proposed by Gravouil, Combescure, Herry & Faucher. In the first part of this paper, we extended modal reduction to subdomains undergoing finite rigid‐body rotations. Here, we focus on the consequences of local modal projection (either linear or geometrically non‐linear) on the treatment of interface problems between subdomains. In particular, we address the issues of the invertibility and efficiency of the solution process. We illustrate our propositions with specific interpretations of the examples presented in Part 1 and present an additional example to demonstrate the properties of special sets of modes. Copyright © 2004 John Wiley & Sons, Ltd.     

An HR-method of mesh refinement for boundary element method

This paper describes a mesh refinement technique for boundary element method in which the number of elements, the size of elements and the element end location are determined iteratively in order to obtain a user specified accuracy. The method uses L₁ norm as a measure of error in the density function and a grading function that ensures that error over each element is the same. The use of grading function along with L₁ norm makes the mesh refinement technique applicable to Direct and Indirect boundary element method formulation for a variety of boundary element method applications. Numerical problems in elastostatics, fracture mechanics, and bending of plate solved using Direct and Indirect method in which the density functions are approximated by Linear Lagrange, Quadratic Lagrange or Cubic Hermite polynomials validate the effectiveness of the proposed mesh refinement technique. © 1998 John Wiley & Sons, Ltd.     

A p‐hierarchical adaptive procedure for the scaled boundary finite element method

This paper describes a p‐hierarchical adaptive procedure based on minimizing the classical energy norm for the scaled boundary finite element method. The reference solution, which is the solution of the fine mesh formed by uniformly refining the current mesh element‐wise one order higher, is used to represent the unknown exact solution. The optimum mesh is assumed to be obtained when each element contributes equally to the global error. The refinement criteria and the energy norm‐based error estimator are described and formulated for the scaled boundary finite element method. The effectivity index is derived and used to examine quality of the proposed error estimator. An algorithm for implementing the proposed p‐hierarchical adaptive procedure is developed. Numerical studies are performed on various bounded domain and unbounded domain problems. The results reflect a number of key points. Higher‐order elements are shown to be highly efficient. The effectivity index indicates that the proposed error estimator based on the classical energy norm works effectively and that the reference solution employed is a high‐quality approximation of the exact solution. The proposed p‐hierarchical adaptive strategy works efficiently. Copyright © 2007 John Wiley & Sons, Ltd.     

A finite element method with mesh‐separation‐based approximation technique and its application in modeling crack propagation with adaptive mesh refinement

This paper presents a FEM with mesh‐separation‐based approximation technique that separates a standard element into three geometrically independent elements. A dual mapping scheme is introduced to couple them seamlessly and to derive the element approximation. The novel technique makes it very easy for mesh generation of problems with complex or solution‐dependent, varying geometry. It offers a flexible way to construct displacement approximations and provides a unified framework for the FEM to enjoy some of the key advantages of the Hansbo and Hansbo method, the meshfree methods, the semi‐analytical FEMs, and the smoothed FEM. For problems with evolving discontinuities, the method enables the devising of an efficient crack‐tip adaptive mesh refinement strategy to improve the accuracy of crack‐tip fields. Both the discontinuities due to intra‐element cracking and the incompatibility due to hanging nodes resulted from the element refinement can be treated at the elemental level. The effectiveness and robustness of the present method are benchmarked with several numerical examples. The numerical results also demonstrate that a high precision integral scheme is critical to pass the crack patch test, and it is essential to apply local adaptive mesh refinement for low fracture energy problems. Copyright © 2014 John Wiley & Sons, Ltd.     

基于细分曲面的快速原型制造研究

针对快速成形制造中生成实体表面光滑度不高的问题,把细分曲面造型技术应用到快速成形制造中。先定义了顶点平坦度概念,然后对蝶形细分模式进行了改进,用此方法可以直接在初始的三角网格模型上提高模型的表面光滑度而避免了由CAD造形系统重新生成的问题,由此提出了一种可以提高生成实体表面光滑度方法。从光滑度分析和大量的实验结果可以看出,该方法可有效地提高实体表面光滑度,并具有较好的稳定性。     

Mechanically based models: Adaptive refinement for B‐spline finite element

This article presents two new methods for adaptive refinement of a B‐spline finite element solution within an integrated mechanically based computer aided engineering system. The proposed techniques for adaptively refining a B‐spline finite element solution are a local variant of np‐refinement and a local variant of h‐refinement. The key component in the np‐refinement is the linear co‐ordinate transformation introduced into the refined element. The transformation is constructed in such a way that the transformed nodal configuration of the refined element is identical to the nodal configuration of the neighbour elements. Therefore, the assembly proceeds as with classic finite elements, while the solution approximation conforms exactly along the inter‐element boundaries. For the h‐refinement, this transformation is introduced into a construction that merges the super element from the finite element world with the hierarchical B‐spline representation from the computational geometry. In the scope of developing sculptured surfaces, the proposed approach supports C⁰ as well as the Hermite B‐spline C¹ continuous shapes. For sculptured solids, C⁰ continuity only is considered in this article. The feasibility of the proposed methods in the scope of the geometric design is demonstrated by several examples of creating sculptured surfaces and volumetric solids. Numerical performance of the methods is demonstrated for a test case of the two‐dimensional Poisson equation. Copyright © 2003 John Wiley & Sons, Ltd.     

Fully implicit finite element method for the modeling of free surface flows with surface tension effect

A new approach based on the use of the Newton and level set methods allows to follow the motion of interfaces with surface tension immersed in an incompressible Newtonian fluid. Our method features the use of a high‐order fully implicit time integration scheme that circumvents the stability issues related to the explicit discretization of the capillary force when capillary effects dominate. A strategy based on a consistent Newton–Raphson linearization is introduced, and performances are enhanced by using an exact Newton variant that guarantees a third‐order convergence behavior without requiring second‐order derivatives. The problem is approximated by mixed finite elements, while the anisotropic adaptive mesh refinements enable us to increase the computational accuracy. Numerical investigations of the convergence properties and comparisons with benchmark results provide evidence regarding the efficacy of the methodology. The robustness of the method is tested with respect to the standard explicit method, and stability is maintained for significantly larger time steps compared with those allowed by the stability condition. Copyright © 2016 John Wiley & Sons, Ltd.     

A bubble‐inspired algorithm for finite element mesh partitioning

This paper presents a bubble‐inspired algorithm for partitioning finite element mesh into subdomains. Differing from previous diffusion BUBBLE and Center‐oriented Bubble methods, the newly proposed algorithm employs the physics of real bubbles, including nucleation, spherical growth, bubble–bubble collision, reaching critical state, and the final competing growth. The realization of foaming process of real bubbles in the algorithm enables us to create partitions with good shape without having to specify large number of artificial controls. The minimum edge cut is simply achieved by increasing the volume of each bubble in the most energy efficient way. Moreover, the order, in which an element is gathered into a bubble, delivers the minimum number of surface cells at every gathering step; thus, the optimal numbering of elements in each subdomain has naturally achieved. Because finite element solvers, such as multifrontal method, must loop over all elements in the local subdomain condensation phase and the global interface solution phase, these two features have a huge payback in terms of solver efficiency. Experiments have been conducted on various structured and unstructured meshes. The obtained results are consistently better than the classical kMetis library in terms of the edge cut, partition shape, and partition connectivity. Copyright © 2012 John Wiley & Sons, Ltd.     

Enrichment and coupling of the finite element and meshless methods

A mixed hierarchical approximation based on finite elements and meshless methods is presented. Two cases are considered. The first one couples regions where finite elements or meshless methods are used to interpolate: continuity and consistency is preserved. The second one enriches a finite element mesh with particles. Thus, there is no need to remesh in adaptive refinement processes. In both cases the same formulation is used, convergence is studied and examples are shown. Copyright © 2000 John Wiley & Sons, Ltd.     

A multi‐level adaptive mesh refinement method for level set simulations of multiphase flow on unstructured meshes

An adaptive mesh refinement (AMR) technique is proposed for level set simulations of incompressible multiphase flows. The present AMR technique is implemented for two‐dimensional/three‐dimensional unstructured meshes and extended to multi‐level refinement. Smooth variation of the element size is guaranteed near the interface region with the use of multi‐level refinement. A Courant–Friedrich–Lewy condition for zone adaption frequency is newly introduced to obtain a mass‐conservative solution of incompressible multiphase flows. Finite elements around the interface are dynamically refined using the classical element subdivision method. Accordingly, finite element method is employed to solve the problems governed by the incompressible Navier–Stokes equations, using the level set method for dynamically updated meshes. The accuracy of the adaptive solutions is found to be comparable with that of non‐adaptive solutions only if a similar mesh resolution near the interface is provided. Because of the substantial reduction in the total number of nodes, the adaptive simulations with two‐level refinement used to solve the incompressible Navier–Stokes equations with a free surface are about four times faster than the non‐adaptive ones. Further, the overhead of the present AMR procedure is found to be very small, as compared with the total CPU time for an adaptive simulation. Copyright © 2016 John Wiley & Sons, Ltd.     

A hybrid adaptive finite element phase-field method for quasi-static and dynamic brittle fracture

The phase-field approach has unique advantages in describing fracture phenomena, which has received extensive attention in the past decade. Nevertheless, the phase-field modeling of fracture is computationally demanding, due to the high temporal-spatial resolution required for crack tracking. In this contribution, a novel hybrid adaptive finite element phase-field method (ha-PFM) is developed to solve brittle fracture problems under quasi-static and dynamic loading. ha-PFM can dynamically track the propagation of the cracks and adaptively refine the meshes based on a novel crack tip identification strategy. Afterward, the refined meshes in the noncrack progression region are reconverted into coarse meshes. This scheme prominently reduces the computational cost, eg, CPU time and memory usage. Unlike the previous adaptive phase-field method, multilevel hybrid triangular and quadrilateral elements were developed to discretize the computational domain, which eliminates hanging nodes and ensures that the meshes in the vicinity of the crack tip are highly isotropic. Several representative benchmarks containing quasi-static and dynamic fracture were reinvestigated with ha-PFM, and its excellent performance is substantiated by comparison with the standard phase-field method and literature results.     

An adaptive finite element framework for fatigue crack propagation

A new finite element (FE) framework for fatigue crack propagation (FCP) analysis is proposed. This framework combines the simplicity of standard industrial FCP analysis with the generality and accuracy of a full FE analysis and can be implemented on a small computer by combining standard existing computational tools. In this way it constitutes an attractive alternative to existing approaches. The framework is based on linear elastic fracture mechanics and on FE mesh adaptation. Some novel features are introduced in several of its steps in order to make it efficient and at the same time reasonably accurate. Various computational aspects of the scheme are discussed. A few two‐dimensional numerical examples involving FCP in thin sheets under plane‐stress conditions are presented to demonstrate the performance of the framework. Some of the numerical results are compared to those of laboratory experiments. Copyright © 2002 John Wiley & Sons, Ltd.     

A hierarchical finite element for geometrically non‐linear vibration of doubly curved,moderately thick isotropic shallow shells

A p‐version, hierarchical finite element for doubly curved, moderately thick, isotropic shallow shells is derived and geometrically non‐linear free vibrations of panels with rectangular planform are investigated. The geometrical non‐linearity is due to large displacements, and the effects of the rotatory inertia and transverse shear are considered. The time domain equations of motion are obtained by applying the principle of virtual work and the d'Alembert's principle. These equations are mapped to the frequency domain by the harmonic balance method, and are finally solved by a predictor–corrector method. The convergence properties of the element proposed and the influence of several parameters on the dynamic response are studied. These parameters are the shell's thickness, the width‐to‐length ratio, the curvature‐to‐width ratio and the ratio between curvature radii. The first and higher order modes are analysed. Some results are compared with results published or calculated using a commercial finite element package. It is demonstrated that with the proposed element low‐dimensional, accurate models are obtained. Copyright © 2002 John Wiley & Sons, Ltd.     

有限元网格修正的自适应分析及其应用

本文在对有限元变量连续条件分析的基础上，将应力误差范数用于计算结果的误差估计，使非结构化网格生成系统与有限元计算有机地结合起来，并将网格单元修正的自适应分析应用于二维应力集中问题的研究，从而实现了有限元最佳化离散，提高了有限元数值求解的可靠性和近似程度。     

Space adaptive finite element methods for dynamic Signorini problems

Space adaptive techniques for dynamic Signorini problems are discussed. For discretisation, the Newmark method in time and low order finite elements in space are used. For the global discretisation error in space, an a posteriori error estimate is derived on the basis of the semi-discrete problem in mixed form. This approach relies on an auxiliary problem, which takes the form of a variational equation. An adaptive method based on the estimate is applied to improve the finite element approximation. Numerical results illustrate the performance of the presented method. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Use of material forces in adaptive finite element methods

The theory of material forces for a hyperelastic material is briefly presented using the translational invariance of the control volume. The theory is derived for the dynamical setting, while the numerical implementation is limited to the static case. The finite element (FE) method is used to solve the standard field equations. After obtaining the solution the material force balance is consistently discretized with FE. As a result of this post-processing discrete material forces are obtained. They are then used to set up an adaptive scheme, in which the magnitude of the material forces acts as an indicator for mesh refinement. Special consideration is given to the boundary, where two different refinement strategies are proposed. The two strategies are compared by studying the refinement process for three examples.     

Mesh refinement strategies without mapping of nonlinear solutions for the generalized and standard FEM analysis of 3‐D cohesive fractures

A robust and efficient strategy is proposed to simulate mechanical problems involving cohesive fractures. This class of problems is characterized by a global structural behavior that is strongly affected by localized nonlinearities at relatively small‐sized critical regions. The proposed approach is based on the division of a simulation into a suitable number of sub‐simulations where adaptive mesh refinement is performed only once based on refinement window(s) around crack front process zone(s). The initialization of Newton‐Raphson nonlinear iterations at the start of each sub‐simulation is accomplished by solving a linear problem based on a secant stiffness, rather than a volume mapping of nonlinear solutions between meshes. The secant stiffness is evaluated using material state information stored/read on crack surface facets which are employed to explicitly represent the geometry of the discontinuity surface independently of the volume mesh within the generalized finite element method framework. Moreover, a simplified version of the algorithm is proposed for its straightforward implementation into existing commercial software. Data transfer between sub‐simulations is not required in the simplified strategy. The computational efficiency, accuracy, and robustness of the proposed strategies are demonstrated by an application to cohesive fracture simulations in 3‐D. Copyright © 2016 John Wiley & Sons, Ltd.     

Solution of Fokker-Planck equation by finite element and finite difference methods for nonlinear systems

The response of a structural system to white noise excitation (deltacorrelated) constitutes a Markov vector process whose transitional probability density function (TPDF) is governed by both the forward Fokker-Planck and backward Kolmogorov equations. Numerical solution of these equations by finite element and finite difference methods for dynamical systems of engineering interest has been hindered by the problem of dimensionality. In this paper numerical solution of the stationary and transient form of the Fokker-Planck (FP) equation corresponding to two state nonlinear systems is obtained by standard sequential finite element method (FEM) using C⁰ shape function and Crank-Nicholson time integration scheme. The method is applied to Van-der-Pol and Duffing oscillators providing good agreement between results obtained by it and exact results. An extension of the finite difference discretization scheme developed by Spencer, Bergman and Wojtkiewicz is also presented. This paper presents an extension of the finite difference method for the solution of FP equation up to four dimensions. The difficulties associated in extending these methods to higher dimensional systems are discussed. This paper is dedicated to Prof R N Iyengar of the Indian Institute of Science on the occasion of his formal retirement.