Isothermal flow simulation of liquid composite molding

This paper proposes a finite element/nodal volume procedure for the isothermal flow simulation of liquid composite molding processes. The formulation and the numerical implementation of the procedure are described. A scheme is introduced to prevent the procedure from possible locking in the flow calculation. The capability and the numerical accuracy of the procedure are investigated through a number of numerical examples.     

Design optimization of non-linear structures with rate-dependent and rate-independent constitutive models

A design optimization process for non-linear structures with history-dependent materials modelled using the endochronic constitutive theory is described. This constitutive model does not use the concept of a yield surface and describes plastic, viscoplastic and viscoelastic behaviour with one set of equations. Therefore, development of the yield surface is not traced in numerical calculations, simplifying the implementation of response and sensitivity analyses. The total Lagrangian formulation incorporating both geometric and material non-linear effects is used. Shape, non-shape and material design parameters are treated simultaneously using the control volume concept, and static and dynamic problems are treated. A simple structure is optimized for several cases of static and impact loading conditions to demonstrate the procedure. Plastic and viscoplastic material behaviour as well as shape and non-shape design parameters are treated in the example problems.     

A meshless numerical method based on the local boundary integral equation (LBIE) to solve linear and non-linear boundary value problems

Meshless methods for solving boundary value problems have been extensively popularized in recent literature owing to their flexibility in engineering applications, especially for problems with discontinuities, and because of the high accuracy of the computed results. A meshless method for solving linear and non-linear boundary value problems, based on the local boundary integral equation method and the moving least squares (MLS) approximation, is discussed in the present article. In the present article, the implementation of the LBIE formulation for linear and non-linear problems with the linear part of the differential operator being the Helmholtz type, is developed. For non-linear problems, the total formulation and a rate formulation are developed for the implementation of the presently proposed method. The present method is a true meshless one, as it does not need domain and boundary elements to deal with the volume and boundary integrals, for linear as well as non-linear problems. The “companion solution” is employed to simplify the present formulation and reduce the computational cost. It is shown that the satisfaction of the essential as well as natural boundary conditions is quite simple, and algorithmically very efficient in the present LBIE approach, even when the non-interpolative MLS approximation is used. Numerical examples are presented for several linear and non-linear problems, for which exact solutions are available. The present method converges fast to the final solution with reasonably accurate results for both the unknown variable and its derivatives in solving non-linear problems. No post processing procedure is required to compute the derivatives of the unknown variable [as in the conventional boundary element method and field/boundary element method, as the solution from the present method, using the MLS approximation, is already smooth enough. The numerical results in these examples show that high rates of convergence with mesh refinement for the Sobolev norms 6·6₀ and 6·6₁ are achievable, and that the values of the unknown variable and its derivatives are quite accurate.     

An adjoint variable method for sensitivity analysis of non-linear elastic systems

An adjoint variable method for design sensitivity analysis of non-linear elastic systems is presented. The method uses domain parameterization and a mutual form of the Hu-Washizu energy principle, and extends results reported in a recent work for linear elastic systems to non-linear elasticity. Non-linearities due to finite deformations and non-linear, hyperelastic constitutive models are considered. In contrast to other methods for non-linear sensitivity analysis, the present formulation can be applied with force, displacement or mixed approximate solution methods. The mutual energy expression used in the adjoint sensitivity derivation is developed from a non-linear extension of the Hu-Washizu energy functional and yields a linear governing equation for the adjoint system. This has important ramifications for the computational cost of a sensitivity analyses of non-linear systems: excluding the cost of determining the response of the system, the cost of a sensitivity analysis for a non-linear system is essentially the same as that for a linear system. Finite element implementation of the resulting sensitivity expressions is discussed, and two numerical examples are presented. The first example involves large deformations of a Mooney-Rivlin body, while the second involves design sensitivity analysis for mixed solution methods.     

An alternative version of the Pian–Sumihara element with a simple extension to non-linear problems

 The stiffness matrix for the Pian–Sumihara element can be obtained in a different way than originally presented in Pian and Sumihara (1984). Instead of getting the element matrix from a hybrid stress formulation with five stress terms one can use a modified Hu–Washizu formulation using nine stress and nine strain terms as well as four enhanced strain terms. Using orthogonal stress and strain functions it becomes possible to obtain the stiffness matrix via sparse Bˉ-matrices so that numerical matrix inversions can be omitted. The advantage of using the mixed variational formulation with displacements, stresses, strains, and enhanced strains is that the extension to non-linear problems is easily achieved since the final computer implementation is very similar to an implementation of a displacement element. Received 31 January 2000     

Volume integral equations in non-linear 3-D magnetostatics

In this paper a discussion of volume integral formulations in three-dimensional non-linear magnetostatics is presented. Integral formulations are examined in connection with Whitney's elements in order to find new approaches. A numerical algorithm based on a formulation implying properly the continuity conditions of magnetic field strength H, i.e. an h-type formulation, is introduced. Results of demanding application problems are shown demonstrating the characteristics of this kind of volume integral approach. In addition, a discussion of the parallelized version of the numerical code based on the h-type approach is presented appended with numerical results illustrating the advantages of combining integral formulations with concurrent computing.     

A boundary element method for steady incompressible thermoviscous flow

A boundary element formulation is presented for moderate Reynolds number, steady, incompressible, thermoviscous flows. The governing integral equations are written exclusively in terms of velocities and temperatures, thus eliminating the need for the computation of any gradients. Furthermore, with the introduction of reference velocities and temperatures, volume modelling can often be confined to only a small portion of the problem domain, typically near obstacles or walls. The numerical implementation includes higher order elements, adaptive integration and multiregion capability. Both the integral formulation and implementation are discussed in detail. Several examples illustrate the high level of accuracy that is obtainable with the current method.     

A perturbation analysis and finite element approximate model for free surface flow over curved beds

A perturbation analysis is developed for 1-D shallow water flow over a curved bed for applications such as spillways. The perturbation approach leads to a new formulation of the problem with associated weak integral statement and approximation using finite elements. The flow may exhibit a hydraulic jump in the downstream regime. An artificial dissipation technique is introduced to stabilize the non-linear problem and suppress numerical oscillations. Numerical results demonstrate the performance of the model and compare it with the steep-slope shallow water formulation corresponding to the model with zero curvature.     

Creep deformation structural analysis using an efficient numerical algorithm

This paper presents a fast numerical algorithm for the implementation of material models for creep into finite element codes. First, an overview of existing algorithms for transient and steady-state creep is given. Next, a new formulation is presented which reduces the constitutive integration to the solution of a scalar non-linear algebraic equation. A solution is shown to exist without the need for subincrementation. Details of the numerical algorithm are then discussed. The paper closes with several numerical examples which illustrate the speed, robustness and accuracy of the proposed procedure as implemented in the Lawrence Livermore National Laboratory finite element codes NIKE2D and NIKE3D.     

Finite element modelling of hardening concrete: application to the prediction of early age cracking for massive reinforced structures

This article presents finite element modelling to predict the early age cracking risk of concrete structures. It is a tool to help practitioners choose materials and construction techniques to reduce the risk of cracking. The proposed model uses original hydration modelling (allowing composed binder to be modelled and hydric consumption to be controlled) followed by a non-linear mechanical model of concrete at early ages involving creep and damage coupling. The article considers hydration effects on this mechanical model, which is based on a non-linear viscoelastic formulation combined with an anisotropic, regularized damage model. Details of the numerical implementation are given in the article and the model is applied successively to a laboratory structure and to a massive structure in situ (experimental wall of a nuclear power plant studied in the framework of the French national research project CEOS.fr).     

Hyper-singular boundary element method for elastoplastic fracture mechanics analysis with large deformation

In this paper a two-dimensional hyper-singular boundary element method for elastoplastic fracture mechanics analysis with large deformation is presented. The proposed approach incorporates displacement and the traction boundary integral equations as well as finite deformation stress measures, and general crack problems can be solved with single-region formulations. Efficient regularization techniques are applied to the corresponding singular terms in displacement, displacement derivatives and traction boundary integral equations, according to the degree of singularity of the kernel functions. Within the numerical implementation of the hyper-singular boundary element formulation, crack tip and corners are modelled with discontinuous elements. Fracture measures are evaluated at each load increment, using the J-integral. Several cases studies with different boundary and loading conditions have been analysed. It has been shown that the new singularity removal technique and the non-linear elastoplastic formulation lead to accurate solutions.     

A comparison between finite volume and switched moving boundary approaches for dynamic vapor compression system modeling

Most work in dynamic heat exchanger modeling for control design can be classified as either a finite volume or a moving boundary formulation. These approaches represent fundamentally different discretization approaches and are often characterized as contrasting accuracy with simulation speed. This work challenges that characterization by validating finite volume and moving boundary heat exchanger models with experimental data from a vapor compression system in order to demonstrate that these approaches are capable of achieving similar levels of accuracy. However, there are differences. The moving boundary model is found to have faster simulation speed, while the finite volume model is more flexible for adaptation to heat exchangers of different physical configuration. The formulation of each modeling approach used in this work is described in detail and techniques to increase simulation speed and avoid numerical issues in implementation are discussed.     

An efficient solver of the saturation equation in liquid composite molding processes

A major issue in Liquid Composite Molding Process (LCM) concerns the reduction of voids formed during the resin filling process. Reducing the void content increases the quality of the composite and improves its mechanical properties. Most of modeling efforts on process simulation of mold filling has been focused on the single phase Darcy’s law, with resin as the only phase, ignoring the formation and transport of voids. The resin flow in a partially saturated region can be characterized as two phase flow through a porous medium. The mathematical formulation of saturation in LCM takes into account the interaction between resin and air as it occurs in a two phase flow. This model leads to the introduction of relative permeabilities as a function of saturation. The modified saturation equation is obtained as a result, which is a non-linear advection-diffusion equation with viscous and capillary phenomena. In this work, a flux limiter technique has been used to solve a modified saturation equation for the LCM process. The implemented algorithm allows a numerical optimization of the injected flow rate which minimizes the micro/macroscopic void formation during mold filling. Some preliminary numerical results are presented here in order to validate the proposed mathematical model and the numerical scheme. This formulation opens up new opportunities to improve LCM flow simulations and optimize injection molds.     

Explicit thickness integration for three-dimensional shell elements applied to non-linear analysis

The problem of multilayered degenerated 3-D shell elements for which the numerical integration is performed for each ply is that of the high generation time in non-linear analysis when the number of plies is important. But these elements give accurate results for thin and moderately thick shells, so in order to reduce the generation time explicit thickness integration is investigated. We first write an expansion of the strain-displacement matrix in power series of the thickness variable in order to obtain explicit expressions of the tangent stiffness matrix and internal force vector, appearing in the non-linear formulation. Explicit expressions of non-linear stiffness matrices are presented, using the explicit integration-first approximation. Simple expressions of several matrices, sub-matrices and vectors appearing in the formulation are given here in order to obtain an important computing-time gain. Next, some numerical validation tests comparing the classical element with numerical thickness integration and this one are discussed to prove validity of this formulation.     

Non-linear boundary element formulation applied to contact analysis using tangent operator

This work presents a non-linear boundary element formulation applied to analysis of contact problems. The boundary element method (BEM) is known as a robust and accurate numerical technique to handle this type of problem, because the contact among the solids occurs along their boundaries. The proposed non-linear formulation is based on the use of singular or hyper-singular integral equations by BEM, for multi-region contact. When the contact occurs between crack surfaces, the formulation adopted is the dual version of BEM, in which singular and hyper-singular integral equations are defined along the opposite sides of the contact boundaries. The structural non-linear behaviour on the contact is considered using Coulomb's friction law. The non-linear formulation is based on the tangent operator in which one uses the derivate of the set of algebraic equations to construct the corrections for the non-linear process. This implicit formulation has shown accurate as the classical approach, however, it is faster to compute the solution. Examples of simple and multi-region contact problems are shown to illustrate the applicability of the proposed scheme.     

Parallel finite element method utilizing the mode splitting and sigma coordinate for shallow water flows

Parallel finite element method for the analysis of quasi-three dimensional shallow water flow is presented. The mode splitting technique and the sigma coordinate (generalized coordinate) are employed to use parallel computers effectively. Parallel implementation of the unstructured grid-based formulation is carried out on the Hitachi parallel-super computer SR2201. The tidal flow of Tokyo Bay is simulated for a numerical example. The speed-up ratio and the efficiency of the parallelization are investigated. The present method is shown to be a useful and powerful tool for the large scale computation of shallow water flows.     

A Laplace-transform-based three-dimensional BEM for poroelasticity

This paper presents the formulation and the numerical implementation of a three-dimensional direct boundary element method for the Biot theory of poroelasticity. To avoid the need of time-stepping and volume integration, the solution is performed in the Laplace transform space. Solution in time is obtained via numerical inversion. Several examples, including the settlement of a rectangular footing and a modified Mandel problem, are examined.     

Boundary element method based on a new second kind integral equation formulation

A boundary integral equation (BIE) formulation for elasticity problems with mixed boundary conditions, proposed by Parton and Perlin (Mathematical Methods of the Theory of Elasticity, Mir, Moscow, 1984), is implemented in this paper using quadratic boundary elements. The formulation is specialised to Stokes flow problems by setting the Poisson ratio to 0·5 in the relevant kernels. The implementation is used to analyse non-trivial three dimensional problems in elasticity and Stokes flows. The results compare well with those obtained by a direct boundary element method. An outline of the extension of the formulation to non-linear problems is also given.     

BEM modeling of saturated porous media susceptible to damage

This paper deals with the numerical analysis of saturated porous media, taking into account the damage phenomena on the solid skeleton. The porous media is taken into poro-elastic framework, in full-saturated condition, based on Biot’s Theory. A scalar damage model is assumed for this analysis. An implicit boundary element method (BEM) formulation, based on time-independent fundamental solutions, is developed and implemented to couple the fluid flow and two-dimensional elastostatic problems. The integration over boundary elements is evaluated using a numerical Gauss procedure. A semi-analytical scheme for the case of triangular domain cells is followed to carry out the relevant domain integrals. The non-linear problem is solved by a Newton-Raphson procedure. Numerical examples are presented, in order to validate the implemented formulation and to illustrate its efficacy.     

Finite element analysis of incompressible viscous flow in a helical pipe

This paper presents an accurate finite element procedure to deal with steady state, fully developed and incompressible viscous flow in helical pipes with arbitrary curvatures and torsions. The full Navier-Stokes equations and continuity equation have been explicitly derived using a non-orthogonal helical coordinate system. To obtain the final simultaneous non-linear algebraic equations, a pressure-velocity finite element formulation is formulated based on the Galerkin Method.The combined influence of finite curvature and finite torsion on the helical flow is studied. The secondary flow patterns and contours of axial velocity of helical flows show the significant distinction with those of toroidal flows. Further, the effect of torsion on flow rates can be neglected.Several numerical examples are presented. Excellent correlations between the computed results and available referenced solutions can be drawn.