Variational formulations for surface tension,capillarity and wetting

The interest in the simulation of flows with significant surface tension effects has grown significantly in recent years. This has been driven by the substantial advances made in the measurement and manufacturing of microscopic systems, since at small length scales surface phenomena are dominant. In this article, surface tension, capillarity and wetting effects are discussed in terms of the virtual–work principle and shape sensitivity, starting from first principles and arriving at variational formulations that are adequate for numerical treatment (by finite elements, for example). To make the exposition self-contained, some elements of differential geometry are recalled using a formulation that is fully in Cartesian coordinates and may thus be more friendly to readers not familiar with covariant derivatives. All necessary results are proved in this Cartesian formulation. Several numerical examples computed with a finite element/level set formulation are used to illustrate this challenging physical problem.     

Stabilized finite element formulation for elastic–plastic finite deformations

This paper presents a stabilized finite element formulation for nearly incompressible finite deformations in hyperelastic–plastic solids, such as metals. An updated Lagrangian finite element formulation is developed where mesh dependent terms are added to enhance the stability of the mixed finite element formulation. This formulation circumvents the restriction on the displacement and pressure fields due to the Babuška–Brezzi condition and provides freedom in choosing interpolation functions in the incompressible or nearly incompressible limit, typical in metal forming applications. Moreover, it facilitates the use of low order simplex elements (i.e. P1/P1), reducing the degrees of freedom required for the solution in the incompressible limit when stable elements are necessary. Linearization of the weak form is derived for implementation into a finite element code. Numerical experiments with P1/P1 elements show that the method is effective in incompressible conditions and can be advantageous in metal forming analysis.     

PolyTop: a Matlab implementation of a general topology optimization framework using unstructured polygonal finite element meshes

We present an efficient Matlab code for structural topology optimization that includes a general finite element routine based on isoparametric polygonal elements which can be viewed as the extension of linear triangles and bilinear quads. The code also features a modular structure in which the analysis routine and the optimization algorithm are separated from the specific choice of topology optimization formulation. Within this framework, the finite element and sensitivity analysis routines contain no information related to the formulation and thus can be extended, developed and modified independently. We address issues pertaining to the use of unstructured meshes and arbitrary design domains in topology optimization that have received little attention in the literature. Also, as part of our examination of the topology optimization problem, we review the various steps taken in casting the optimal shape problem as a sizing optimization problem. This endeavor allows us to isolate the finite element and geometric analysis parameters and how they are related to the design variables of the discrete optimization problem. The Matlab code is explained in detail and numerical examples are presented to illustrate the capabilities of the code.     

Predicting residual stresses in steady-state forming processes

A two-dimensional, Eulerian finite element formulation for modeling isotropic, elasto-viscoplastic, steady-state deformations which is capable of predicting residual stresses is presented in this paper. This problem is solved in two parts, namely, solution of the boundary value problem by a mixed finite element formulation for the velocity and pressure fields, and integration of the constitutive equations along pathlines across the domain. In this formulation, a discontinuous pressure field is used in the finite element formulation to reduce the system of equations to a system for only the velocity field. A new method for integrating the constitutive equations is also presented which improves the efficiency of the algorithm.     

Shape design sensitivity analysis of kinematical boundaries

A new approach is used in this paper to derive the design sensitivity formulation with kinematical design boundaries. By employing the concept of the conventional finite difference approach, the variation of structural response due to change of the kinematic design boundary can be represented by the perturbed structure under a set of kinematical boundary conditions. Parameterization of the design variation with respect to the design variable enables us to transform the design sensitivity into the solutions of a boundary value problem with perturbation displacements on the design boundary. The perturbation diplacements can be evaluated from the stress and displacement fields of the initial problem. This approach can be treated as a special case of the general direct formulation, but the derivation using the finite difference procedure gives a strong physical meaning of the method, and the formulation derived provides an explicit form for design sensitivity calculation. The numerical implementation of this approach based on the boundary element method is discussed, and a few numerical examples are used to verify the proposed formulation.     

Post-buckling behaviour of elastic circular plates using a simple finite element formulation

The post-buckling behaviour of elastic circular plates is studied in this paper using a simple finite element formulation. The final linearized eigenvalue problem is solved by using three numerical methods and all these methods are found to yield accurate results for moderately large deflections. Comparison of the present results with the results existing in literature shows the validity of this formulation.     

SSD: Smooth Signed Distance Surface Reconstruction

We introduce a new variational formulation for the problem of reconstructing a watertight surface defined by an implicit equation, from a finite set of oriented points; a problem which has attracted a lot of attention for more than two decades. As in the Poisson Surface Reconstruction approach, discretizations of the continuous formulation reduce to the solution of sparse linear systems of equations. But rather than forcing the implicit function to approximate the indicator function of the volume bounded by the implicit surface, in our formulation the implicit function is forced to be a smooth approximation of the signed distance function to the surface. Since an indicator function is discontinuous, its gradient does not exist exactly where it needs to be compared with the normal vector data. The smooth signed distance has approximate unit slope in the neighborhood of the data points. As a result, the normal vector data can be incorporated directly into the energy function without implicit function smoothing. In addition, rather than first extending the oriented points to a vector field within the bounding volume, and then approximating the vector field by a gradient field in the least squares sense, here the vector field is constrained to be the gradient of the implicit function, and a single variational problem is solved directly in one step. The formulation allows for a number of different efficient discretizations, reduces to a finite least squares problem for all linearly parameterized families of functions, and does not require boundary conditions. The resulting algorithms are significantly simpler and easier to implement, and produce results of quality comparable with state‐of‐the‐art algorithms. An efficient implementation based on a primal‐graph octree‐based hybrid finite element‐finite difference discretization, and the Dual Marching Cubes isosurface extraction algorithm, is shown to produce high quality crack‐free adaptive manifold polygon meshes.     

Vibration of thin cylindrical shells based on a mixed finite element formulation

A Hellinger-Reissner functional for thin circular cylindrical shells is presented. A mixed finite element formulation is developed from this functional, which is free from line integrals and relaxed continuity terms. This element is applied to the problem of vibration of rectangular cylindrical shells. Bilinear trial functions are used for all field variables. The element satisfies the compatibility and completeness requirements. The numerical results obtained in this work show that convergence is quite rapid and monotonic for a much smaller number of degrees of freedom than other finite element formulations.     

Stabilized low-order finite elements for frictional contact with the extended finite element method

Contact problem suffers from a numerical instability similar to that encountered in incompressible elasticity, in which the normal contact pressure exhibits spurious oscillation. This oscillation does not go away with mesh refinement, and in some cases it even gets worse as the mesh is refined. Using a Lagrange multipliers formulation we trace this problem to non-satisfaction of the LBB condition associated with equal-order interpolation of slip and normal component of traction. In this paper, we employ a stabilized finite element formulation based on the polynomial pressure projection (PPP) technique, which was used successfully for Stokes equation and for coupled solid-deformation–fluid-diffusion using low-order mixed finite elements. For the frictional contact problem the polynomial pressure projection approach is applied to the normal contact pressure in the framework of the extended finite element method. We use low-order linear triangular elements (tetrahedral elements for 3D) for both slip and normal pressure degrees of freedom, and show the efficacy of the stabilized formulation on a variety of plane strain, plane stress, and three-dimensional problems.     

A reduced integration finite element technology based on a thermomechanically consistent stabilisation for 3D problems

In this paper we suggest a new finite element technology for thermomechanically fully coupled problems. It is based on the method of reduced integration with hourglass stabilisation. The proposed formulation allows the evaluation of the additional thermal field at one Gauss point, e.g. in the centre of the element. One crucial aspect is the Taylor expansion of all constitutively dependent variables, as e.g. the heat flux, the internal and the external rates of dissipation, with respect to the centre of the element. In this way a so-called thermal hourglass stabilisation, analogously to the classical mechanical hourglass stabilisation, is derived. The thermal stabilisation parameters are defined well and the computational efficiency which comes along with the consistent formulation is very high. The new element formulation is applied on thermomechanically coupled problems of finite elastoplasticity. It can be also easily used in the context of other multi-field problems.     

Geometrically non-linear formulation for the three dimensional solid-shell transition finite elements

This paper presents a geometrically non-linear formulation using total lagrangian approach for the solid-shell transition finite elements. Such transition finite elements are necessary in geometrically non-linear analysis of structures modelled with three dimensional solid elements and the curved shell elements. These elements are an essential connecting link between the solid elements and the shell elements. The element formulation presented here is derived using the properties of the three dimensional solid elements and the curved shell elements. No restrictions are imposed on the magnitude of the nodal rotations. Thus the element formulation is capable of handling large rotations between two successive load increments. The element properties are derived and presented in detail. Numerical examples are also presented to demonstrate their behavior, accuracy and applications in three dimensional stress analysis.

It is shown that the selection of different stress and strain components at the integration points do not effect the overall linear response of the element. However, in geometrically non-linear applications it may be necessary to select appropriate stress and the strain components at the integration points for stable and converging element behavior. Numerical examples illustrate various characteristics of the element.     

A new numerical approach to a non-steady filtration problem

A free-boundary transient problem of seepage flow is studied from a numerical standpoint. From a suitable formulation of the problem in terms of variational inequality we introduce a new numerical approach of the implicit type and based on the finite element method. In this approach the problem is solved on a fixed region and the position of the free boundary is automatically found as part of the solution of the problem; so it is not necessary to solve a succession of problems with different positions of the free boundary. We prove stability and convergence for the approximate solution and we give several numerical results. Work supported by C. N. R. of Italy through the Laboratorio di Analisi Numerica of Pavia.     

Local and global error estimations in linear structural dynamics

The study discusses the concept of error estimation in linear elastodynamics. Two different types of error estimators are presented. First ‘classical’ methods based on post-processing techniques are discussed starting from a semidiscrete formulation. The temporal error due to the finite difference discretization is measured independently of the spatial error of the finite element discretization. The temporal error estimators are applied within one time step and the spatial error estimators at a time point. The error is measured in the global energy norm. The temporal evolution of the error cannot be reflected. Furthermore the estimators can only evaluate the mean error of the whole spatial domain. As the second scheme local error estimators are presented. These estimators are designed to evaluate the error of local variables in a certain region by applying duality techniques. Local estimators are known from linear elastostatics and have later on been extended to nonlinear problems. The corresponding dual problem represents the influence of the local variable on the initial problem and may be related to the reciprocal theorem of Betti–Maxwell. In the present study this concept is transferred to linear structural dynamics. Because the dual problem is established over the total space–time domain, the spatial and temporal error of all time steps can be accumulated within one procedure. In this study the space–time finite element method is introduced as a single field formulation.     

On various formulations of large amplitude free vibrations of beams

Various finite element formulations of large amplitude free vibrations of beams with immovably supported ends are discussed in this paper. Analytical formulation based on the Rayleigh-Ritz method is also presented. Numerical results of the analytical approach are seen to be in good agreement with some of these finite element formulations. Mixed finite element formulations based on two methods are derived to study the large amplitude free vibrations of beams. The mixed finite element methods also show good agreement with the analytical and the above finite element formulations. Various points of view raised from time to time on the applicability of these formulations can now be clarified through these formulations and the numerical results. The weakness of the so-called improved Ritz-type finite element model in predicting the nonlinear frequency ratio is highlighted through various results of the above formulations. As a typical example, a hinged-hinged beam on immovable ends is considered for all the above formulations and the nonlinear frequencies show a good agreement amongst themselves at all amplitude levels.     

Finite element method and limit analysis theory for soil mechanics problems

This paper deals with a finite element formulation of problems of limit loads in soil mechanics via limit analysis theory. After recalling the principal results of this theory, the authors describe a numerical formulation for both the static and kinematic approaches of the ultimate load. Thanks to linearization of the yield criterion, the finite element model leads to a linear programming problem. The efficiency of the two proposed computing procedures is demonstrated by their application to the problem of pulling out of foundations and slope stability.     

Wave-induced oscillations in harbours of variable depth

An integral formulation of the Helmboltz equation is numerically tested on the problem of the wave-induced oscillations in harbours of variable depth. If the method has some disadvantages with respect to the finite element methods, it is rather simple to apply and in some cases it might be preferred. An accurate analysis of all its features and possibilities has not been carried out. A simple application has been considered. The results obtained for this problem show that the volume of the harbour basin, when the bottom is varied, is of primary importance. As the volume decreases, resonance wavenumbers decrease and peak amplitudes increase.     

On minimization of stress concentration for three-dimensional models

A solution procedure is described for the minimization of stress concentration in general three-dimensional bodies. The formulation is based on tetrahedral finite element analysis and linear programming optimization. Analytical sensitivity analysis, omitting the use of repeated analyses, is presented.Two examples are described in detail. Firstly, a three-dimensional recalculation of the plane stress/plane strain optimal elliptical shape demonstrates the reliability of the procedure, since extremely good agreement is found. Secondly, the three-dimensional cavity problem is treated. The known solution of an ellipsoid is found to an accuracy governed by the finite element mesh.This example demonstrates the need for a better finite element model, and the paper therefore finally focuses on the problem of model-optimization (best finite element model) of a given design. It is shown that the formulation of this optimal remodelling problem is parallel to that of the optimal redesign problem. A remodelling criterion based on combined global/local accuracy is suggested.