An adaptive remeshing procedure for discontinuous finite element limit analysis

An adaptive remeshing procedure is proposed for discontinuous finite element limit analysis. The procedure proceeds by iteratively adjusting the element sizes in the mesh to distribute local errors uniformly over the domain. To facilitate the redefinition of element sizes in the new mesh, the interelements discontinuous field of elemental bound gaps is converted into a continuous field, ie, the intensity of bound gap, using a patch‐based approximation technique. An analogous technique is subsequently used for the approximation of element sizes in the old mesh. With these information, an optimized distribution of element sizes in the new mesh is defined and then scaled to match the total number of elements specified for each iteration in the adaptive remeshing process. Finally, a new mesh is generated using the advancing front technique. This adaptive remeshing procedure is repeated several times until an optimal mesh is found. Additionally, for problems involving discontinuous boundary loads, a novel algorithm for the generation of fan‐type meshes around singular points is proposed explicitly and incorporated into the main adaptive remeshing procedure. To demonstrate the feasibility of our proposed method, some classical examples extracted from the existing literary works are studied in detail.     

Updated Lagrangian free surface flow simulations with natural neighbour Galerkin methods

In this paper, a new method to simulate free surface fluid flows within an updated Lagrangian framework is described. It is based on the use of a meshless technique coined as natural element method (NEM) or, more recently, as natural neighbour Galerkin method. The position of the flow front or the geometry of the fluid domain is handled by invoking the geometrical concept of α‐shape of the cloud of points, thus avoiding the explicit definition of the boundary of the domain as it evolves. This method also avoids the traditional need of remeshing typical in finite element simulations of this kind of processes. Three types of fluid behaviour have been considered, namely a purely Newtonian fluid, a non‐Newtonian short fibre‐reinforced thermoplastic, and finally a Norton–Hoff viscoplastic behaviour. Benchmark examples showing the performance of the technique are included in the paper. Copyright © 2004 John Wiley & Sons, Ltd.     

The extended finite element method for rigid particles in Stokes flow

A new method for the simulation of particulate flows, based on the extended finite element method (X‐FEM), is described. In this method, the particle surfaces need not conform to the finite element boundaries, so that moving particles can be simulated without remeshing. The near field form of the fluid flow about each particle is built into the finite element basis using a partition of unity enrichment, allowing the simple enforcement of boundary conditions and improved accuracy over other methods on a coarse mesh. We present a weak form of the equations of motion useful for the simulation of freely moving particles, and solve example problems for particles with prescribed and unknown velocities. Copyright © 2001 John Wiley & Sons, Ltd.     

On finite element modelling of surface tension: Variational formulation and applications – Part II: Dynamic problems

In Part I, a finite element model of surface tension has been discussed and used to solve some quasi-static problems. The quasi-static analysis is often required to find not only the initial shape of the liquid but also the static equilibrium state of a liquid body before a dynamic analysis can be carried out. In general, natural and industrial processes in which surface tension force is dominant are of dynamic nature. In this second part of this work, the dynamic effects will be included in the finite element model described in Part I.A fully Lagrangian finite element method is used to solve the free surface flow problem and Newtonian constitutive equations describing the fluid behaviour are approximated over a finite time interval. As a result the momentum equations are function of nodal position instead of velocities. The resulting ordinary differential equation is integrated using Newmark algorithm. To avoid overly distorted elements an adaptive remeshing strategy is adopted. The adaptive strategy employs a remeshing indicator based on viscous dissipation functional and incorporates an appropriate transfer operator.The validation of the model is performed by comparing the finite element solutions to available analytical solutions of a droplet oscillations and experimental results pertaining to stretching of a liquid bridge.     

Unification of parametric and implicit methods for shape sensitivity analysis and optimization with fixed mesh

Parametric and implicit methods are traditionally thought to be two irrelevant approaches in structural shape optimization. Parametric method works as a Lagrangian approach and often uses the parametric boundary representation (B‐rep) of curves/surfaces, for example, Bezier and B‐splines in combination with the conformal mesh of a finite element model, while implicit method relies upon level‐set functions, that is, implicit functions for B‐rep, and works as an Eulerian approach in combination with the fixed mesh within the scope of extended finite element method or finite cell method. The original contribution of this work is the unification of both methods. First, a new shape optimization method is proposed by combining the features of the parametric and implicit B‐reps. Shape changes of the structural boundary are governed by parametric B‐rep on the fixed mesh to maintain the merit in computer‐aided design modeling and avoid laborious remeshing. Second, analytical shape design sensitivity is formulated for the parametric B‐rep in the framework of fixed mesh of finite cell method by means of the Hamilton–Jacobi equation. Numerical examples are solved to illustrate the unified methodology. Copyright © 2016 John Wiley & Sons, Ltd.     

An adaptive boundary‐element approach for 3D transient free surface cavity flow,as applied to polymer processing

An adaptive (Lagrangian) boundary‐element approach is proposed for the general three‐dimensional simulation of confined free surface flow of viscous incompressible fluids. The method is stable as it includes remeshing capabilities of the deforming free surface, and thus can handle large deformations. A simple algorithm is developed for mesh refinement of the deforming free surface mesh. Smooth transition between large and small elements is achieved without significant degradation of the aspect ratio of the elements in the mesh. Several flow problems are presented to illustrate the utility of the approach, particularly as encountered in polymer processing. These problems illustrate the transient nature of the flow in the extrusion through circular and square dies, the filling of circular and square disks as in conventional injection molding, and the flow during gas‐assisted injection molding inside a duct, with relevance to the important problem of viscous fingering. Copyright © 2001 John Wiley & Sons, Ltd.     

FINITE ELEMENT REMESHING: A METAL FORMING APPROACH FOR QUADRILATERAL MESH GENERATION AND REFINEMENT

The paper presents an automatic finite element remeshing system for quadrilateral elements consisting of modules for mesh generation, densification, smoothing and interpolation of field variables. The mesh generator takes into account the contour of the old mesh, eventual interference with dies and the plastic deformation of the material. An initial coarse mesh is created by utilizing a grid-based approach for creating well-shaped internal elements, in conjunction with a nodal connection approach based on constrained Delaunay triangulation, for linking with the boundary. Subsequent local mesh refinement is performed according to parameters depending on past, present and predicted future deformation related field variables; being, respectively, the strain gradient and strain rate distribution in relation with the velocity field, element size and quality. Smoothing is accomplished using an iterative Laplacian repositioning method. As illustrated in the presented examples this overall strategy ensures a robust and efficient remeshing scheme for finite element simulation of bulk metal-forming processes. © 1997 by John Wiley & Sons, Ltd.     

Simulation of two‐ and three‐dimensional viscoplastic flows using adaptive mesh refinement

This paper presents a finite element solver for the simulation of steady non‐Newtonian flow problems, using a regularized Bingham model, with adaptive mesh refinement capabilities. The solver is based on a stabilized formulation derived from the variational multiscale framework. This choice allows the introduction of an a posteriori error indicator based on the small scale part of the solution, which is used to drive a mesh refinement procedure based on element subdivision. This approach applied to the solution of a series of benchmark examples, which allow us to validate the formulation and assess its capabilities to model 2D and 3D non‐Newtonian flows.     

Particulate flow simulations using lubrication theory solution enrichment

A technique for the numerical simulation of suspensions of particles in fluid based on the extended finite element method (X‐FEM) is developed. In this method, the particle surfaces need not conform to the finite element boundaries, so that moving particles can be simulated without remeshing. The finite element basis is enriched with the Stokes flow solution for flow past a single particle and the lubrication theory solution for flow between particles. The latter enrichment allows the simulation of particles that come arbitrarily close together without refining the mesh in the gap between them. Example problems illustrating both types of enrichment are shown, along with a study of a 50% solution in channel flow. Copyright © 2003 John Wiley & Sons, Ltd.     

A Lagrangian finite element approach for the analysis of fluid–structure interaction problems

A Lagrangian finite element method for the analysis of incompressible Newtonian fluid flows, based on a continuous re‐triangulation of the domain in the spirit of the so‐called Particle Finite Element Method, is here revisited and applied to the analysis of the fluid phase in fluid–structure interaction problems. A new approach for the tracking of the interfaces between fluids and structures is proposed. Special attention is devoted to the mass conservation problem. It is shown that, despite its Lagrangian nature, the proposed combined finite element‐particle method is well suited for large deformation fluid–structure interaction problems with evolving free surfaces and breaking waves. The method is validated against the available analytical and numerical benchmarks. Copyright © 2010 John Wiley & Sons, Ltd.     

LLM and X-FEM based interface modeling of fluid–thin structure interactions on a non-interface-fitted mesh

This paper presents a non-interface-fitted mesh method for fluid–thin structure interactions. The key components are the Lagrangian Lagrange-multiplier (LLM) method and the extended finite element method (X-FEM). The LLM couples fluid and thin structure through the Lagrangian nodes of the structure element. The X-FEM gives flow discontinuity to the fluid elements intersected by the structure element. The combination method is verified through applications to flow with a domain-partitioning boundary and flow-induced flapping of a flexible filament. We discuss how the discontinuities at the interface enhance the simulation results, how the lack of the discontinuities affects the results, and identify some effects of these discontinuity enrichments.     

An adaptive boundary-element approach to 3D transient free-surface flow of viscous fluids

An adaptive (Lagrangian) boundary-element approach is proposed for the general three-dimensional simulation of confined free-surface flow of viscous incompressible fluids. The method is stable as it includes remeshing capabilities of the deforming free surface, and thus can handle large deformations. A simple algorithm is developed for mesh refinement of the deforming free surface mesh. Smooth transition between large and small elements is achieved without significant degradation of the aspect ratio of the elements in the mesh. Several flow problems are presented to illustrate the utility of the approach.     

An immersed finite element method with integral equation correction

We propose a robust immersed finite element method in which an integral equation formulation is used to enforce essential boundary conditions. The solution of a boundary value problem is expressed as the superposition of a finite element solution and an integral equation solution. For computing the finite element solution, the physical domain is embedded into a slightly larger Cartesian (box‐shaped) domain and is discretized using a block‐structured mesh. The defect in the essential boundary conditions, which occurs along the physical domain boundaries, is subsequently corrected with an integral equation method. In order to facilitate the mapping between the finite element and integral equation solutions, the physical domain boundary is represented with a signed distance function on the block‐structured mesh. As a result, only a boundary mesh of the physical domain is necessary and no domain mesh needs to be generated, except for the non‐boundary‐conforming block‐structured mesh. The overall approach is first presented for the Poisson equation and then generalized to incompressible viscous flow equations. As an example of fluid–structure coupling, the settling of a heavy rigid particle in a closed tank is considered. Copyright © 2010 John Wiley & Sons, Ltd.     

Explicit Reproducing Kernel Particle Methods for large deformation problems

The explicit Reproducing Kernel Particle Method (RKPM) is presented and applied to the simulations of large deformation problems. RKPM is a meshless method which does not need a mesh structure in its formulation. Because of this mesh-free property, RKPM is able to simulate large deformation problems without remeshing which is often required for the mesh-based methods such as the finite element method. The RKPM shape function and its derivatives are constructed by imposing the consistency conditions. An efficient treatment of essential boundary conditions is also proposed for explicit time integration. The Lagrangian method based on the reference configuration is employed for the RKPM simulation of large deformation problems. Several examples of non-linear elastic materials are solved to demonstrate the performance of the method. The numerical experiment for the problem of underwater bubble explosion is also performed using the explicit Lagrangian RKPM formulation. © 1998 John Wiley & Sons, Ltd.     

Adaptive moving mesh level set method for structure topology optimization

The level set method is a promising approach to provide flexibility in dealing with topological changes during structural optimization. Normally, the level set surface, which depicts a structure's topology by a level contour set of a continuous scalar function embedded in space, is interpolated on a fixed mesh. The accuracy of the boundary positions is therefore largely dependent on the mesh density, a characteristic of any Eulerian expression when using a fixed mesh. This article combines the adaptive moving mesh method with a level set structure topology optimization method. The finite element mesh automatically maintains a high nodal density around the structural boundaries of the material domain, whereas the mesh topology remains unchanged. Numerical experiments demonstrate the effect of the combination of a Lagrangian expression for a moving mesh and a Eulerian expression for capturing the moving boundaries.     

Immersed smoothed finite element method for fluid–structure interaction simulation of aortic valves

This paper presents a novel numerical method for simulating the fluid?Cstructure interaction (FSI) problems when blood flows over aortic valves. The method uses the immersed boundary/element method and the smoothed finite element method and hence it is termed as IS-FEM. The IS-FEM is a partitioned approach and does not need a body-fitted mesh for FSI simulations. It consists of three main modules: the fluid solver, the solid solver and the FSI force solver. In this work, the blood is modeled as incompressible viscous flow and solved using the characteristic-based-split scheme with FEM for spacial discretization. The leaflets of the aortic valve are modeled as Mooney-Rivlin hyperelastic materials and solved using smoothed finite element method (or S-FEM). The FSI force is calculated on the Lagrangian fictitious fluid mesh that is identical to the moving solid mesh. The octree search and neighbor-to-neighbor schemes are used to detect efficiently the FSI pairs of fluid and solid cells. As an example, a 3D idealized model of aortic valve is modeled, and the opening process of the valve is simulated using the proposed IS-FEM. Numerical results indicate that the IS-FEM can serve as an efficient tool in the study of aortic valve dynamics to reveal the details of stresses in the aortic valves, the flow velocities in the blood, and the shear forces on the interfaces. This tool can also be applied to animal models studying disease processes and may ultimately translate to a new adaptive methods working with magnetic resonance images, leading to improvements on diagnostic and prognostic paradigms, as well as surgical planning, in the care of patients.     

Finite‐element analysis of a combined fine‐blanking and extrusion process

This paper presents the characteristics of the combined fine‐blanking and extrusion process and gives a detailed analysis of the process with the finite‐element method. To carry out the simulation step by step and avoid the tendency to diverge in the calculations, the remeshing, tracing and golden section methods were developed and introduced into the finite‐element program. Different boundary conditions were used in the simulation; the mesh distortion, field of material flow, and the stress and strain distributions were obtained. From the simulated results, the deformation characteristics under different boundary conditions were revealed. An experiment was also carried out to verify the simulated results. A large strain analysis technique was chosen to determine the effective strain distribution based on the experiment. The effective strain distributions from the simulation are in accordance with the effective strain distributions and the hardness distributions from the experiment. Copyright © 2005 John Wiley & Sons, Ltd.     

Arbitrary Lagrangian–Eulerian finite element analysis of strain localization in transient problems

Non-local models guaranty that finite element computations on strain softening materials remain sound up to failure from a theoretical and computational viewpoint. The non-locality prevents strain localization with zero global dissipation of energy, and consequently finite element calculations converge upon mesh refinements to non-zero width localization zones. One of the major drawbacks of these models is that the element size needed in order to capture the localization zone must be smaller than the internal length. Hence, the total number of degrees of freedom becomes rapidly prohibitive for most engineering applications and there is an obvious need for mesh adaptivity. This paper deals with the application of the arbitrary Lagrangian–Eulerian (ALE) formulation, well known in hydrodynamics and fluid–structure interaction problems, to transient strain localization in a non-local damageable material. It is shown that the ALE formulation which is employed in large boundary motion problems can also be well suited for non-linear transient analysis of softening materials where localization bands appear. The remeshing strategy is based on the equidistribution of an indicator that quantifies the interelement jump of a selected state variable. Two well known one-dimensional examples illustrate the capabilities of this technique: the first one deals with localization due to a propagating wave in a bar, and the second one studies the wave propagation in a hollow sphere.     

A p-adaptive scaled boundary finite element method based on maximization of the error decrease rate

This study enhances the classical energy norm based adaptive procedure by introducing new refinement criteria, based on the projection-based interpolation technique and the steepest descent method, to drive mesh refinement for the scaled boundary finite element method. The technique is applied to p-adaptivity in this paper, but extension to h- and hp-adaptivity is straightforward. The reference solution, which is the solution of the fine mesh formed by uniformly refining the current mesh, is used to represent the unknown exact solution. In the new adaptive approach, a projection-based interpolation technique is developed for the 2D scaled boundary finite element method. New refinement criteria are proposed. The optimum mesh is assumed to be obtained by maximizing the decrease rate of the projection-based interpolation error appearing in the current solution. This refinement strategy can be interpreted as applying the minimisation steepest descent method. Numerical studies show the new approach out-performs the conventional approach.