Standard Finite Element Analysis Research Articles

Viscoelastic Material Models of Polypropylene for Thermoforming Applications

Polypropylene (PP), a semi-crystalline material, is typically solid phase thermoformed at temperatures associated with crystalline melting, generally in the 150° to 160°Celsius range. In this very narrow thermoforming window the mechanical properties of the material rapidly decline with increasing temperature and these large changes in properties make Polypropylene one of the more difficult materials to process by thermoforming. Measurement of the deformation behaviour of a material under processing conditions is particularly important for accurate numerical modelling of thermoforming processes. This paper presents the findings of a study into the physical behaviour of industrial thermoforming grades of Polypropylene. Practical tests were performed using custom built materials testing machines and thermoforming equipment at Queen′s University Belfast. Numerical simulations of these processes were constructed to replicate thermoforming conditions using industry standard Finite Element Analysis software, namely ABAQUS and custom built user material model subroutines. Several variant constitutive models were used to represent the behaviour of the Polypropylene materials during processing. This included a range of phenomenological, rheological and blended constitutive models. The paper discusses approaches to modelling industrial plug-assisted thermoforming operations using Finite Element Analysis techniques and the range of material models constructed and investigated. It directly compares practical results to numerical predictions. The paper culminates discussing the learning points from using Finite Element Methods to simulate the plug-assisted thermoforming of Polypropylene, which presents complex contact, thermal, friction and material modelling challenges. The paper makes recommendations as to the relative importance of these inputs in general terms with regard to correlating to experimentally gathered data. The paper also presents recommendations as to the approaches to be taken to secure simulation predictions of improved accuracy.

Modelling well leakage in multilayer aquifer systems using the extended finite element method

The extended finite element method (XFEM) is applied to the problem of predicting the steady-state leakage from layered sedimentary aquifer systems perforated by abandoned wells. Multi-aquifer systems are modelled using a quasi-three-dimensional model where the head distribution in each aquifer is assumed to be two-dimensional. The standard finite element method is locally enriched in the vicinity of both injection and abandoned wells so that the logarithmic singularity in the solution at the wells is more accurately approximated. Two versions of the discrete equations are developed. In the first, no assumptions are made about the relative alignment of the meshes in adjacent aquifers and a relatively complex, but general, system of equations results. In the second, the meshes in adjacent aquifers are assumed to be aligned and system of equations is less complex. In the simplified case, the model can be broken down into a set of three building blocks—aquifer elements, aquitard elements and well elements. Numerical examples show that the standard Finite Element Method (FEM) poorly approximates the head and the flux rate at the abandoned wells and that the convergence rate, in terms of leakage through the abandoned wells, of the FEM is on the order of O( h 0.4). In contrast, the XFEM is shown to be more than two orders of magnitude more accurate for coarse meshes. Furthermore, the XFEM solution converges four times faster, at a rate of about O( h 1.8). The excellent computational efficiency and flexibility of the XFEM model makes it an attractive alternative to both the standard finite element analysis and to semi-analytical methods for predicting well leakage in multi-aquifer systems.

Mathematical Assessment of the Effects of Parabolic and Spherical Surface Topographies on the Interfacial State of Stress

A mathematical procedure to utilize the complementary energy method was developed, by minimization, in order to find an approximate analytical solution to the 3-D stress distributions in bonded interfaces of dissimilar materials. In order to incorporate the effects of surface topography, the interface was expressed as a general surface in Cartesian coordinates, i.e., F (x, y, z) = 0. The 3-D stress functions were used to produce 3-D stress components in dissimilar materials. At the interface, the internal tractions in each of the coordinate directions were balanced by the mathematical procedure. By using a penalty function method of the optimization theory, integration of the complementary energy produced the necessary equations to solve the 3-D stress distribution problem at the interface. A noticeable advantage of our method is that the stress jumps at the interface predicted in elasticity theory are captured, while standard finite element analysis (FEA) methods usually have difficulty to show such stress jumps at interfaces. The 3-D mathematical procedure we developed for obtaining the stress components at bonded bi-material interfaces offers significant promise in solving interface problems with different surface topographies, which can be described mathematically. Thus, the procedure developed provides an efficient tool to optimize interface construction by various methods such as chemical (etching), mechanical (machining, roughening, etc.), and other novel methods, such as laser ablation, currently becoming available to achieve desired interface stress distributions for bonded materials. In this paper, the parabolic interface problem, i.e., y = x2, and the spherical interface problem, i.e., y = x2/4 + z2/4, are considered for an aluminum/epoxy interface.

Nonlinear Algebraic Reduction for Snap-Fit Simulation

Snap-fits are often preferred over other traditional methods of assembly since they reduce part count, assembly/disassembly time, and manufacturing cost. A typical simulation of a snap-fit assembly entails two critical steps: contact detection and nonlinear deformation analysis, which are strongly coupled. One could rely on standard 3D contact detectors and standard 3D finite element analysis (FEA) to simulate the two steps. However, since snap-fits are slender, 3D FEA tends to be inefficient and slow. On the other hand, one could rely on an explicit 1D beam reduction for simulation. This is highly efficient for large deformation analysis, but contact detection can pose challenges. Moreover, for complex snap-fits, extracting cross-sectional properties for 1D formulation is nontrivial. We propose here a nonlinear algebraic dimensional reduction method that offers the “best of both worlds.” In the proposed method, a 3D model is used for contact detection. However, to speed-up deformation analysis, the 3D model is implicitly reduced to a 1D beam model via an algebraic process. Thus, it offers the generality of 3D simulation and the computational efficiency of 1D simulation, as confirmed by the numerical experiments and optimization studies.

SEA modelling for sound package development — inclusion of simulated airborne noise loads

During the last years, the statistical energy analysis (SEA) has been widely used by vehicle manufacturers and suppliers to analyse the airborne noise on road vehicles. While the basic workflow for the construction of vehicle SEA models is now well established, an accepted specific methodology for the evaluation of the SEA airborne loads to be used in these models is not available yet. In the framework of a joint project between Rieter Automotive and Volkswagen, a hybrid procedure that combines the standard finite element analysis (FEA) with the recently developed energy boundary element analysis (EBEA) was used for the evaluation of such loads and validated obtaining rather satisfactory results.

Nosing of thin-walled PVC tubes into hollow spheres using a die

In a recent work, the authors presented a new single-stage nosing process for manufacturing metallic hollow spheres from thin-walled tubes that is capable of extending the formability limits of conventional nosing (Alves and Martins in J Strain Anal Eng Des 43:205–216, 2008). The fundamentals of the process were investigated by means of standard finite element analysis and its technological viability was evaluated by means of experimentation on industrial AA6060 aluminium alloy tubes. This paper is aimed on extending the scope of single-stage nosing so as to include the production of polymer hollow spheres. The investigation draws from the development of an innovative extension of the flow formulation that is capable of modeling cold plastic deformation of pressure-sensitive yield surfaces under a nonassociated flow rule to the identification and analysis of the major process parameters that influence formability and quality of the polymer hollow spheres. The overall performance of the process is assessed by means of a combined theoretical and experimental investigation made with industrial polyvinyl chloride (PVC) tubes. Results show that the innovative extension of the finite element flow formulation is capable of successfully modeling cold forming of polymers and that single-stage cold nosing of thin-walled PVC tubes can be successfully utilized for manufacturing sound hollow spheres.

Free vibration analysis of ribbed plates by a combined analytical–numerical method

An easy to code and computationally efficient analytical–numerical method for quick prediction of the modal characteristics of rectangular ribbed plates is presented. The approach is suitable for low-frequency free vibration analysis of thin rectangular plates reinforced by a small number of light stiffeners. The assumed-modes method is used to formulate the equations of motion of the plate and the rib separately. The motion of the built-up structure is then obtained by enforcing appropriate continuity conditions between the two. The resulting sparse generalized eigenvalue problem can be profitably solved by reliable methods to calculate only a small set of selected eigenmodes. An alternative formulation is also proposed for the single-ribbed case which leads to a compact analytical form of the frequency equation whose solution can be easily determined either graphically or numerically. The present method demonstrates good agreement with published results and standard finite element analysis.

Finite element analysis of the thermoforming of Polypropylene

Polypropylene (PP) is typically solid phase thermoformed at temperatures close to its crystalline melting point, usually in the 150° to 160° Celsius range. In such conditions the mechanical properties of the material rapidly decline with temperature and these large changes in properties make Polypropylene one of the more difficult materials to process by thermoforming. This paper presents the findings of a study into the thermoforming behaviour of an industrial thermoforming grade of Polypropylene. Practical tests were performed using custom built thermoforming equipment at Queen’s University Belfast. Numerical simulations of these processes were similarly constructed to replicate thermoforming processes using industry standard Finite Element Analysis software.

A finite element based stability test for equilibria of flexible structures in circular orbits

Large flexible structures could be one of the most challenging space technologies of the future. For example, the well-known space elevator may one day guide astronauts along a 36,000 km long tether from the earth's surface to a geostationary orbit. In this paper, we imagine an arbitrary space structure in a stationary circular motion around a celestial body. Usually such systems require two problems to be solved: the strength of the materials and an overall stability analysis. The latter task is a non-trivial one since in orbiting systems not the angular rate but the angular momentum must be kept constant when applying, for example, Dirichlet's criterion. In particular, this fact becomes important if the system's dimension is in the magnitude of the orbital radius. Therefore, this paper will focus on a general applicable method to determine the stability of flexible earth-orbiting structures. The analysis process is based on the reduced energy momentum method presented by J.C. Simo, T.A. Posbergh and J.E. Marsden, which has been customized for use in systems with cyclic coordinates by the author. In the presented approach the second variation of an amended potential is derived from a standard finite element analysis. An efficient stability test is introduced, which limits the computational effort to the evaluation of a few eigenvalues and eigenvectors of the system's tangent stiffness matrix. Finally, two examples are discussed to demonstrate the application of the method.

On standard and vector finite element analysis of a strict anti-plane shear plasticity model with elastic curvature

A finite deformation plasticity model with plasticity driven by curvature of lattice elastic deformation (due to geometrically necessary dislocations (GNDs)) and by scalar, statistically averaged lattice elastic deformation (due to statistically stored dislocations (SSDs)), is formulated for strict anti-plane shear kinematics and implemented by standard and vector finite element (FE) methods. The curl of the elastic part of the deformation gradient represents the elastic lattice curvature that a grain or subgrain cell experiences as a result of the presence of GNDs at grain or subgrain cell boundaries. “Strict” anti-plane shear implies that the form of the elastic and plastic parts of the deformation gradient are the same as the total deformation gradient, which is the unity tensor plus the dyadic product of the out-of-plane normal and displacement gradient. The plasticity model with elastic curvature leads to a partial differential equation (PDE) to solve for the plastic vector, rather than an ordinary differential equation for standard plasticity without curvature. Along with the linear momentum balance, this leads to a coupled system of PDEs for which to solve numerically by a coupled FE implementation. Aside from the out-of-plane displacements, a plastic vector is solved for at the nodes for the standard element, and along element edges for the vector element. Numerical examples verify the standard and vector element implementations, and demonstrate the additional hardening due to the elastic curvature generated at the notched crack tip for a mode III anti-plane shear crack simulation.

Modeling of Multilayered Structures on the Coupling Surface Between a Structure and an Acoustic Cavity

This paper describes a new numerical method to calculate efficiently the effect of a multilayered structure which is located on the coupling surface between structure and acoustic cavity. This method utilizes the transfer matrix built from analytical wave function sets for each layer. Solutions for Helmholtz equations derived from Biot's theory are applied for the layer of absorbing porous material such as polymer foams. Transfer matrix for multilayered structure is then applied to form boundary conditions between structure and acoustic cavity and the modified equations of motions for the coupled systems are obtained. The new calculation method employed transfer matrix for multilayered structure is finally validated by a numerical example which resulted into the reduction of calculation time by the factor up to 40 without the severe loss of accuracy comparing results by the standard finite element analysis.

A differential geometry approach to analysis of thermal forming

Many products, such as ship hulls, require that metal plates or sheets be formed into complex curvatures. Line heating is a technique commonly used to thermo-mechanically form desired curvatures into plate. The in-plane and bending strain induced by application of a heat line on a plate are embodied in the fundamental coefficients of the resultant shape. It is proposed that the fundamental coefficients from a single heat scan can be superimposed to represent a complex, multi-line heating pattern. An optimization algorithm can then be employed to estimate the resultant shape. This technique is developed and used to provide a method to analyze the thermal forming process based on differential geometry that is more computationally efficient than standard large deformation finite element analysis (FEA) simulations. First, the groundwork for this approach is laid by briefly reviewing the mathematical description of formed surfaces based on differential geometry. The fundamental coefficients of surfaces generated through large-deformation moving-source FEA line-heating simulations are compared at different mesh densities, to gain insight into the physical interpretation of the fundamental coefficients, specifically as applied to the line heating problem. This comprises a suitable framework in which to introduce a method to utilize differential geometry for analysis of deformations caused by line heating patterns. Finally, the practicality of the technique is evaluated as the structural rigidity of the plate is increased through application of multiple heat lines through comparison to large deformation FEA simulations. The new technique exhibits significant computational savings compared to conventional simulation methods.

Multi-objective meta level soft computing-based evolutionary structural design

Evolutionary structural design has been the topic of much recent research; however, such designs are usually hampered by the time-consuming stage of prototype evaluations using standard finite element analysis (FEA). Replacing the time-consuming FEA by neural network approximations may be a computationally efficient alternative, but the error in such approximation may misguide the optimization procedure. In this paper, a multi-objective meta-level (MOML) soft computing-based evolutionary scheme is proposed that aims to strike a balance between accuracy vs. computational efficiency and exploration vs. exploitation. The neural network (NN) is used here as a pre-filter when fitness is estimated to be of lesser significance while the standard FEA is used for solutions that may be optimal in their current population. Furthermore, a fuzzy controller updates parameters of the genetic algorithm (GA) in order to balance exploitation vs. exploration in the search process, and the multi-objective GA optimizes parameters of the membership functions in the fuzzy controller. The algorithm is first optimized on two benchmark problems, i.e. a 2-D Truss frame and an airplane wing. General applicability of the resulting optimization algorithm is then tested on two other benchmark problems, i.e. a 3-layer composite beam and a piezoelectric bimorph beam. Performance of the proposed algorithm is compared with several other competing algorithms, i.e. a fuzzy-GA–NN, a GA–NN, as well as a simple GA that only uses only FEA, in terms of both computational efficiency and accuracy. Statistical analysis indicates the superiority as well as robustness of the above approach as compared with the other optimization algorithms. Specifically, the proposed approach finds better structural designs more consistently while being computationally more efficient.

Design methodology to investigate contact fatigue damage in turbine engine hardware

One of the most significant maintenance drivers in current turbine engines is contact fatigue damage where turbine blades are inserted into the disk. This is commonly referred to as the dovetail slot. The stress state at the dovetail slot is difficult to determine with standard finite element analysis because of the nonlinear nature of the contact problem. Because of this, a design methodology based on singular integral equation analysis has been developed. This design methodology is used for the first time to compare the contact stress behavior of similar components from two different turbine engines. The use of singular integral equation methods to analyze these components is supported by a similar analysis on a dovetail experimental fixture. Stress predictions by the singular integral equation method compare well to finite element methods for static stress conditions in a dovetail slot. Additionally, the results from the component analysis indicate that there is a correlation between stress state and damage seen during maintenance. However, details concerning the correlation still need to be resolved.

Improving the computational efficiency in finite element analysis of shells with uncertain properties

This work is concerned with improving the computational efficiency of the most time consuming tasks performed in Monte Carlo simulation-based Finite Element Analysis (FEA) of shell structures with uncertain properties. For this purpose, stochastic field values are generated on a coarse mesh and then interpolated onto the fine mesh used for the standard FEA computations; the cost-effective TRIC shell element is used to ensure the formation of stiffness matrices in reasonable processing times; the solution of finite element equations is efficiently handled with hybrid schemes combining both iterative and direct solution concepts; additional computational gains are achieved with the use of parallel computing through the straightforward partitioning of the overall Monte Carlo simulation process. The adoption of such advanced computational approaches allows simulation-based probabilistic or stochastic FEA of shells to be performed in affordable computing times and therefore become more tractable in structural engineering practice. The computational procedures described in this work are evaluated on a cluster of 16 networked PCs using three linear elastic test problems with uncertain material and/or geometric parameters: (a) the Scordelis-Lo shell, (b) a pinched cylinder and (c) a 3D steel frame discretized with shell elements.

Analysis of anisotropic prismatic sectiosn

The dynamic behaviour of rotor blades is often modelled using onedimensional beam analysis with equivalent mass and stiffness properties to those of the full blade. Calculation of accurate elastic stiffness terms for these arbitrarily shaped sections with differing material properties is vital to this process. A method which produces these properties using standard finite element analysis codes is presented. The method is then compared with theoretical results for a simple rectangular section beam and case studies are performed on a composite laminate and box-section.

Multi-objective and multi-loading optimization of ultralightweight truss materials

A multi-parameter optimization procedure for ultralightweight truss-core sandwich panels is presented in this paper. To satisfy high design requirements, the proposed approach incorporates objectives from various structural performances of the material system at many loading cases simultaneously. To determine an optimal topology of the truss-core panel, the procedure combines standard finite element (FE) analysis with structural system profile analysis and multi-factor optimization techniques. Detailed configuration and sizing for both facesheets and individual struts in the sandwich panel are optimized. Two examples are presented: (i) tetragonal core panel, and (ii) plagihedral pyramidal core panel. The optimization improves the structural performance of each panel under multiple loading cases and minimizes its structural mass simultaneously.

THEORETICAL DETERMINATION OF ANISOTROPIC EFFECTIVE THERMAL CONDUCTIVITY IN TRANSFORMER WINDINGS

A two-dimensional region representing a typical transformer winding cross-section is studied analytically and numerically. The winding region is characterized by layers of round or flattened conductors separated in one direction by insulating paper. The voids within the windings are filled with transformer oil. The theoretical limits of the effective thermal conductivity in two directions are determined using both the parallel isotherms and the parallel adiabats approaches. A steady state conduction analysis in the composite region is performed using standard commercial finite element analysis software. The conductor curvature radius and contact widths were varied relative to the paper thickness for a typical value of oil thermal conductivity and for aluminum and copper conductors. It was found that the effective thermal conductivity was strongly affected by the paper and that there was negligible difference between results for aluminum and copper conductors. It was observed that the ratio of the effective thermal conductivities in the two directions could be 300 in some cases.

Numerical methods for the topology optimization of structures that exhibit snap‐through

AbstractThe inclusion of non‐linear elastic analyses into the topology optimization problem is necessary to capture the finite deformation response, e.g. the geometric non‐linear response of compliant mechanisms. In previous work, the non‐linear response is computed by standard non‐linear elastic finite element analysis. Here, we incorporate a load–displacement constraint method to traverse non‐linear equilibrium paths with limit points to design structures that exhibit snap‐through behaviour. To accomplish this, we modify the basic arc length algorithm and embed this analysis into the topology optimization problem. Copyright © 2002 John Wiley & Sons, Ltd.

The behaviour of damage tolerant hat-stiffened composite panels loaded in uniaxial compression

Damage tolerant hat-stiffened thin-skinned composite panels with and without a centrally located circular cutout, under uniaxial compression loading, were investigated experimentally and analytically. These panels incorporated a highly postbuckling design characterised by two integral stiffeners separated by a large skin bay with a high width to skin-thickness ratio. In both configurations, the skin initially buckled into three half-wavelengths and underwent two mode-shape changes; the first a gradual mode change characterised by a central deformation with double curvature and the second a dynamic snap to five half-wavelengths. The use of standard path-following non-linear finite element analysis did not consistently capture the dynamic mode change and an approximate solution for the prediction of mode-changes using a Marguerre-type Rayleigh-Ritz energy method is presented. Shortcomings with both methods of analysis are discussed and improvements suggested. The panels failed catastrophically and their strength was limited by the local buckling strength of the hat stiffeners.