A finite element formulation based on the Cosserat point theory

The theory of Cosserat points is the basis of a 3D finite element formulation for large deformations in structural mechanics, that recently was presented by [1]. First investigations [2] have revealed, that this formulation is free of showing undesired locking or hourglassing-phenomena. It additionally shows excellent behaviour for any type of incompressible material, for large deformations and sensitive structures such as plates or shells. The formulation initially was restricted to a Neo-Hookean material. This work will present the extension to a general elastic Ogden material and the verification of the chosen model. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A finite element formulation based on the theory of a Cosserat point

The theory of Cosserat points is the basis of a 3D finite element formulation allowing for large deformations in structural mechanics, that recently was presented by [1]. First attempts have revealed, that this formulation is free of showing undesired locking or hourglassing-phenomena. It additionally shows excellent behaviour for any type of incompressible material, for large deformations and sensitive structures such as plates or shells. Within the theory of Cosserat points, the position vectors X and x , are described through director vectors D _i and d _i by use of trilinear shape functions Nⁱ for an 8-node brick element. The special choice of shape functions Nⁱ allows for director vectors with which the deformation can be split into a homogeneous and an inhomogeneous part. This split enables the use of stiffnesses that correspond to different deformation modes. Analytical solutions to the inhomogeneous deformation modes are incorporated in the formulation and avoid the undesired phenomena. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Buckling of laminated plates - A simple finite element based on higher-order theory

A four-noded rectangular element with seven degrees of freedom at each node is developed for buckling analysis of laminated plate structures having any number of layers with a constant thickness of individual layers. The displacement model is so chosen that it can explain adequately the parabolic distribution of transverse shear stresses and the non-linearity of in-plane displacements across the thickness. A geometrical stiffness matrix is developed using in-plane stresses. A wide range of plates from thick to thin are examined under uniaxial loading conditions. The results are compared with the existing analytical and numerical solutions. The present formulations confirm its applicability for buckling analysis of a wide range of plates.     

Estimation of temperature in rubber-like materials using non-linear finite element analysis based on strain history

A finite element procedure for hyper-elastic materials such as rubber has been developed to estimate the temperature rise during cyclic loading. The irreversible mechanical work developed in rubber has been used to determine the heat generation rate for carrying out thermal analysis. The evaluation of the heat energy is dependent on the strains. The finite element analysis assumes Green–Lagrangian strain displacement relations, Mooney–Rivlin strain energy density function for constitutive relationship, incremental equilibrium equations, and Total Lagrangian approach and the stress and strain of the rubber-like materials are evaluated using a degenerated shell element with assumed strain field technique, considering both material and geometric non-linearities. A transient heat conduction analysis has been carried out to estimate the temperature rise for different time steps in rubber-like materials using Galerkin's formulations. A numerical example is presented and the computed temperature values for various load steps agree closely with the experimental results reported in the literature.     

A finite element analysis of effects of ultrasonic welding parameters on the temperature distribution in wire bonding

Wire bonding is an essential process in the automotive industry. Multi-strand flexible aluminium cables are used for connection of different electronic components and electrical centres in cars. As an alternative for crimping technology in wire bonding, ultrasonic welding (USW) is applied, which is a rapid manufacturing process used to create solid joints between mating materials at low energy consumption compared to the known welding processes, such as oxy-fuel welding and arc welding. An ultrasonic welding machine consists of different parts, such as pneumatic cylinder, piezoelectric converter, booster, welding sonotrode and anvil. Despite of the simplicity of the USW process, choosing the right machine and process parameters, like pressure of the pneumatic cylinder, welding time as well as vibration amplitude of the piezo-converter, is a tricky and complicated task for obtaining an adequate bond. Experimental investigations done in this area are extremely time-consuming and require a lot of effort. Therefore, some new approaches must be developed to understand the process in more detail. The present study focuses on the influence of the ultrasonic welding parameters, such as sonotrode pressure and vibration amplitude, on the temperature distribution at interfaces of two mating pieces in wire bonding [1,2]. Investigations are done by means of FEM simulations as well as by experiments. The results are then extended to thermo-mechanical analysis of multi-strand models. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A finite element method for the solution of a potential theory integral equation

This paper discusses a finite element approximation for an integral equation of the second kind deduced from a potential theory boundary value problem in two variables. The equation is shown to admit a unique solution, to be variational and coercive in the Hilbert space of functions σ ε H^1/2(Γ), f_rd γ = 0. The Galerkin method with finite elements as trial functions is shown to lead to an optimal rate of convergence.     

An extended beam theory for smart materials applications part I: Extended beam models,duality theory,and finite element simulations

An extended theory for elastic and plastic beam problems is studied. By introducing new dependent and independent variables, the standard Timoshenko beam model is extended to take account of shear variation in the lateral direction. The dynamic governing equations are established via Hamilton's principle, and existence and uniqueness results for the solution of the static problem are proved. Using the theory of convex analysis, the duality theory for the extended beam model is developed. Moreover, the extended theory for rigid-perfectly plastic beams is also established. Based on the extended model, a finite-element method is proposed and numerical results are obtained indicating the usefulness of the extended theory in applications.The work of the first author was supported in part by National Science Foundation under Grant DMS9400565.     

A finite element model of temperature distribution in the human torso

A finite element model is developed for the calculation of steady state temperature distribution throughout the human torso. The torso is considered as a cylinder and the differential equations of the model are expressed in their equivalent variational form. The solution is approximated using a rational finite element basis which fully exploits symmetry.     

A non-conforming finite element for limit analysis of periodic composites

In this work, a non-conforming three-dimensional finite element coupled with direct methods and homogenization technique is presented for the limit analysis of periodic metal-matrix composites. Using this element, which is constructed from bilinear shape functions and enriched by internal second-order polynomials, limit analysis of composite material can be efficiently carried out. Accuracy and overall performance are illustrated through comparison with different structural solid elements in the context of direct as well as incremental methods. It is shown that the limit domain of periodic composites for different fiber distributions and volume fractions provides a foundation for the structural design. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A monotonicity analysis theory for the dependent eigenvalues of the vibrating systems

This paper presents and discusses the monotonicity analysis theory for the generalized eigenvalues of the nonlinear structural eigensystems. The analysis is based on investigating the mass and stiffness matrices which are associated with both the mixed and exact finite element models. These models can be distinguished by the shape functions derived from the choice of displacement field which plays a crucial role in the accuracy and efficiency of the solution. The main emphasis of this contribution is the derivation and the investigation of this analysis for large scale eigenproblems.     

A comparative analysis of adaptive algorithms in the finite element method for solving the boundary value problem for a stationary reaction-diffusion equation

A new adaptive algorithm is proposed for constructing grids in the hp-version of the finite element method with piecewise polynomial basis functions. This algorithm allows us to find a solution (with local singularities) to the boundary value problem for a one-dimensional reaction-diffusion equation and smooth the grid solution via the adaptive elimination and addition of grid nodes. This algorithm is compared to one proposed earlier that adaptively refines the grid and deletes nodes with the help of an estimate for the local effect of trial addition of new basis functions and the removal of old ones. Results are presented from numerical experiments aimed at assessing the performance of the proposed algorithm on a singularly perturbed model problem with a smooth solution.     

Modeling the nonlinear deformation of composite laminates based on plasticity theory

The plasticity theory has been successfully used for describing the nonlinear deformation of laminated composite materials under a monotonically increasing loading. Generally, several tests are needed to determine the parameters of the plastic potential for a laminate. We explore an alternative approach and obtain the plastic potential by using theoretical considerations based on a laminate analysis. The model is shown to provide an accurate prediction for the response of a cross-ply glass/epoxy laminate under uniaxial tensile loading at different angles to the material orthotropy axes. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 43, No. 3, pp. 309–318, May–June, 2007.     

A novel far-boundary condition for the finite element analysis of infinite reservoir

The present paper deals with the finite element analysis of the reservoir of infinite extent using a novel far-boundary condition. The equations of motion are expressed in terms of the pressure only assuming water as inviscid and incompressible. The truncation boundary condition is developed numerically from the classical wave equation. Comparative studies show that the proposed far-boundary condition is numerically efficient and accurate over the existing ones, available in the literature. The effect of the geometry of the reservoir bed and the adjacent structure on the development hydrodynamic pressure has been studied. The results show that the geometry of the reservoir bed and as well as the adjacent structure has considerable effect on the development of hydrodynamic pressure at the dam–reservoir interface.     

A new finite element model for the analysis of arbitrary stiffened shells

A new stiffened shallow shell finite element has been introduced for the static analysis of stiffened plates and shells. This approach has been presented to cater for stiffeners in which the positions and properties remain undisturbed in the formulation and the element can accomodate the stiffener anywhere within the shell element and in any direction, which introduces a considerable flexibility in the analysis. This is a distinct improvement over the existing models. Stiffened shells having various disposition of stiffeners as available in the literature, have been analysed by the proposed approach. Comparison obtained with the existing theoretical and/or experimental values have indicated good accuracy with relatively coarser mesh sizes and less CPU time.     

A unifying theory of a posteriori error control for nonconforming finite element methods

Residual-based a posteriori error estimates were derived within one unifying framework for lowest-order conforming, nonconforming, and mixed finite element schemes in Carstensen [Numer Math 100:617–637, 2005]. Therein, the key assumption is that the conforming first-order finite element space annulates the linear and bounded residual ℓ written . That excludes particular nonconforming finite element methods (NCFEMs) on parallelograms in that . The present paper generalises the aforementioned theory to more general situations to deduce new a posteriori error estimates, also for mortar and discontinuous Galerkin methods. The key assumption is the existence of some bounded linear operator with some elementary properties. It is conjectured that the more general hypothesis (H1)–(H3) can be established for all known NCFEMs. Applications on various nonstandard finite element schemes for the Laplace, Stokes, and Navier–Lamé equations illustrate the presented unifying theory of a posteriori error control for NCFEM. Supported by DFG Research Center MATHEON “Mathematics for key technologies” in Berlin and the German Indian Project DST-DAAD (PPP-05). J. Hu was partially supported by National Science Foundation of China under Grant No.10601003.     

A least-squares finite element method based on the Helmholtz decomposition for hyperbolic balance laws

In this paper, a least-squares finite element method for scalar nonlinear hyperbolic balance laws is proposed and studied. The approach is based on a formulation that utilizes an appropriate Helmholtz decomposition of the flux vector and is related to the standard notion of a weak solution. This relationship, together with a corresponding connection to negative-norm least-squares, is described in detail. As a consequence, an important numerical conservation theorem is obtained, similar to the famous Lax–Wendroff theorem. The numerical conservation properties of the method in this paper do not fall precisely in the framework introduced by Lax and Wendroff, but they are similar in spirit as they guarantee that when L² convergence holds, the resulting approximations approach a weak solution to the hyperbolic problem. The least-squares functional is continuous and coercive in an H⁻¹-type norm, but not L²-coercive. Nevertheless, the L² convergence properties of the method are discussed. Convergence can be obtained either by an explicit regularization of the functional, that provides control of the L² norm, or by properly choosing the finite element spaces, providing implicit control of the L² norm. Numerical results for the inviscid Burgers equation with discontinuous source terms are shown, demonstrating the L² convergence of the obtained approximations to the physically admissible solution. The numerical method utilizes a least-squares functional, minimized on finite element spaces, and a Gauss–Newton technique with nested iteration. We believe that the linear systems encountered with this formulation are amenable to multigrid techniques and combining the method with adaptive mesh refinement would make this approach an efficient tool for solving balance laws (this is the focus of a future study).     

A stabilized mixed finite element method for the biharmonic equation based on biorthogonal systems

We propose a stabilized finite element method for the approximation of the biharmonic equation with a clamped boundary condition. The mixed formulation of the biharmonic equation is obtained by introducing the gradient of the solution and a Lagrange multiplier as new unknowns. Working with a pair of bases forming a biorthogonal system, we can easily eliminate the gradient of the solution and the Lagrange multiplier from the saddle point system leading to a positive definite formulation. Using a superconvergence property of a gradient recovery operator, we prove an optimal a priori estimate for the finite element discretization for a class of meshes.     

A partitioned finite element scheme based on Gauge-Uzawa method for time-dependent MHD equations

In this paper, we mainly introduce a partitioned scheme based on Gauge-Uzawa finite element method for the 2D time-dependent incompressible magnetohydrodynamics (MHD) equations. It is a fully decoupled projection method which combines the Gauge and Uzawa methods within a variational formulation. Firstly, the temporal discretization is based on backward Euler technique for the linear term and semi-implicit scheme for the nonlinear term. Secondly, the spatial approximation of fluid velocity, hydrodynamic pressure, and magnetic field apply the mixed element method. Finally, the validity, reliability, and accuracy of the proposed algorithms are supported by numerical experiments.     

A finite element method for a biharmonic equation based on gradient recovery operators

A new non-conforming finite element method is proposed for the approximation of the biharmonic equation with clamped boundary condition. The new formulation is based on a gradient recovery operator. Optimal a priori error estimates are proved for the proposed approach. The approach is also extended to cover a singularly perturbed problem.