Special-purpose Element Research Articles

A preliminary study of the concentration of special purpose elements and precious metals in the enriched precipitates produced by a wet method of cathode-ray tube recycling

The purpose of the work is to investigate the existence and the possibility of recovering economically interesting elements, such as precious metals and rare earth elements, from phosphor layers of cathode ray tube wastes. Therefore, a series of analyses took place in the precipitation material that resulted from the disassembling and wet processing of the cathode-ray tube screens. An applied recycling method, that mechanically with water enriches precipitates of cathode ray tube glass (wet cathode ray tube recycling method), is hosted in a small-medium recycling company. Identification of the amorphous and crystalline phases in combination with the micro-structural characterization and the chemical, major, and trace-element analysis were performed in waste materials as well as in the precipitate that resulted from the wet cathode-ray tube recycling process. Important quantities of zirconium, yttrium, lanthanum, cerium, and gold appear in the precipitate. Conclusively, a hydraulic classification enrichment method should be, furthermore, examined.

Numerical Testing and Validation of LVI on Composite Plates

Laminated composite plates are widely used in the aerospace field, the prediction of their residual life is a interesting challenge for research communities. Their structural behaviour could be affected by several rupture mechanisms due to exercise loading conditions. One of the most critical is the low velocities impacts with different impact energies. This paper deals with an experimental test program performed in order to validate a numerical model developed by using finite element method. All experimental tests were carried out under international standard ASTM D7136 while all numerical tests were carried out by use of a worldwide numerical code Abaqus®. Inter-laminar and intra-laminar rupture mechanisms were taken into account and special-purpose elements were used. Rupture criteria were implemented in the numerical models thanks to their functional ease; results of numerical-experimental comparison were presented and discussed.

Special-purpose elements to impose Periodic Boundary Conditions for multiscale computational homogenization of composite materials with the explicit Finite Element Method

A novel methodology is presented to introduce Periodic Boundary Conditions (PBC) on periodic Representative Volume Elements (RVE) in Finite Element (FE) solvers based on dynamic explicit time integration. This implementation aims at overcoming the difficulties of the explicit FE method in dealing with standard PBC. The proposed approach is based on the implementation of a user-defined element, named a Periodic Boundary Condition Element (PBCE), that enforces the periodicity between periodic nodes through a spring-mass-dashpot system. The methodology is demonstrated in the multiscale simulation of composite materials. Two showcases are presented: one at the scale of computational micromechanics, and another one at the level of computational mesomechanics. The first case demonstrates that the proposed PBCE allows the homogenization of composite ply properties through the explicit FE method with increased efficiency and similar reliability with respect to the equivalent implicit simulations with traditional PBC. The second case demonstrates that the PBCE coupled with Periodic Laminate Elements (PLE) can effectively be applied to the computational homogenization of elastic and strength properties of entire laminates taking into account highly nonlinear effects. Both cases motivate the application of the methodology in multiscale virtual testing in support of the building-block certification of composite materials.

Numerical investigation of onset and evolution of LVI damages in Carbon–Epoxy plates

In order to study the onset and the evolution of low velocity impact damages in Carbon–Epoxy plates, a numerical investigation has been led. A detailed finite element model has been created by using the finite element code Abaqus® which, thanks to the different implemented algorithms, allowed considering both intra-laminar and inter-laminar failure criteria.In particular, the numerical modelling technique of such failure criteria allowed predicting delamination growth, by using special purpose-elements (cohesive elements) and fiber and matrix failure, by using Hashin criteria.Moreover, with the aim to reduce the required CPU time, a global/local finite element modelling approach has been proposed.For validation purpose, numerical results have been compared with data from two sessions of experimental impact tests. The considered impact energy values are 6J, 10J and 13J respectively.

Dynamic crack propagation simulation with scaled boundary polygon elements and automatic remeshing technique

An efficient methodology for automatic dynamic crack propagation simulations using polygon elements is developed in this study. The polygon mesh is automatically generated from a Delaunay triangulated mesh. The formulation of an arbitrary n-sided polygon element is based on the scaled boundary finite element method (SBFEM). All kind of singular stress fields can be described by the matrix power function solution of a cracked polygon. Generalised dynamic stress intensity factors are evaluated using standard finite element stress recovery procedures. This technique does not require local mesh refinement around the crack tip, special purpose elements or nodal enrichment functions. An automatic local remeshing algorithm that can be applied to any polygon mesh is developed in this study to accommodate crack propagation. Each remeshing operation involves only a small patch of polygons around the crack tip, resulting in only minimal change to the global mesh structure. The increase of the number of degrees-of-freedom caused by crack propagation is moderate. The method is validated using four dynamic crack propagation benchmarks. The predicted dynamic fracture parameters show good agreement with experiment observations and numerical simulations reported in the literature.

Numerical implementation of local effects due to two-dimensional discontinuous loads using special elements based on boundary integrals

Abstract In this paper, special-purpose elements are developed for solving local effects caused by discontinuous loads such as concentrated forces, line loads and patch loads applied in plane elastic structures. During the derivation of the special-purpose elements, the interior displacement and stress fields are composed of two parts: (1) the homogeneous solution part, which is represented by a linear combination of fundamental solutions at a number of source points outside the element domain; and (2) the particular load-dependent part, which is analytically represented by suitable local solutions. Meanwhile the independent frame displacements defined over the element boundary are approximated by conventional shape functions. The linkage between the two independent fields is established through use of a newly constructed hybrid variational functional, in which discontinuous loads are treated as generalized body forces. Using the property of delta function, the domain integral associated with discontinuous loads in the variational functional can be removed. The advantage of such special-purpose elements is that a large element, independent of the location of discontinuous loads, can be used to avoid the requirement of mesh refinement in the vicinity of the area with local loads. Numerical experiments are carried out to verify the special-purpose elements and to investigate their effectiveness in terms of mesh reduction and accuracy.

Application of trefftz BEM to anti-plane piezoelectric problem

Anti-plane electroelastic problems are studied by the Trefftz boundary element method (BEM) in this paper. The Trefftz BEM is based on a weighted residual formulation and indirect boundary approach. In particular the point-collocation and Galerkin techniques, in which the basic unknowns are the retained expansion coefficients in the system of complete equations, are considered. Furthermore, special Trefftz functions and auxiliary functions which satisfy exactly the specified boundary conditions along the slit boundaries are also used to derive a special purpose element with local defects. The path-independent integral is evaluated at the tip of a crack to determine the energy release rate for a mode III fracture problem. In final, the accuracy and efficiency of the Trefftz boundary element method are illustrated by an example and the comparison is made with other methods.

Trefftz Finite Element Method and Its Applications

This paper presents an overview of the Trefftz finite element and its application in various engineering problems. Basic concepts of the Trefftz method are discussed, such as T-complete functions, special purpose elements, modified variational functionals, rank conditions, intraelement fields, and frame fields. The hybrid-Trefftz finite element formulation and numerical solutions of potential flow problems, plane elasticity, linear thin and thick plate bending, transient heat conduction, and geometrically nonlinear plate bending are described. Formulations for all cases are derived by means of a modified variational functional and T-complete solutions. In the case of geometrically nonlinear plate bending, exact solutions of the Lamé-Navier equations are used for the in-plane intraelement displacement field, and an incremental form of the basic equations is adopted. Generation of elemental stiffness equations from the modified variational principle is also discussed. Some typical numerical results are presented to show the application of the finite element approach. Finally, a brief summary of the approach is provided and future trends in this field are identified. There are 151 references cited in this revised article.

A practical large-strain solid finite element for sheet forming

An alternative approach for developing practical large-strain finite elements has been introduced and used to create a three-dimensional solid element that exhibits no locking or hourglassing, but which is more easily and reliably derived and implemented than typical reduced-integration schemes with hourglassing control. Typical large-strain elements for forming applications rely on reduced integration to remove locking modes that occur with the coarse meshes that are necessary for practical use. This procedure introduces spurious zero-energy deformation modes that lead to hourglassing, which in turn is controlled by complex implementations that involve lengthy derivations, knowledge of the material model, and/or undetermined parameters. Thus, for a new material or new computer program, implementation of such elements is a daunting task. Wang–Wagoner-3-dimensions (WW3D), a mixed, hexahedral, three-dimensional solid element, was derived from the standard linear brick element by ignoring the strain components corresponding to locking modes while maintaining full integration (8 Gauss points). Thus, WW3D is easily implemented for any material law, with little chance of programming error, starting from programming for a readily available linear brick element. Surprisingly, this approach and resulting element perform similarly or better than standard solid elements in a series of numerical tests appearing in the literature. The element was also tested successfully for an applied sheet-forming analysis problem. Many variations on the scheme are also possible for deriving special-purpose elements. Copyright © 2005 John Wiley & Sons, Ltd.

Earthquake-induced interaction between adjacent reinforced concrete structures with non-equal heights

The influence of the structural pounding on the ductility requirements and the seismic behaviour of reinforced concrete structures designed to EC2 and EC8 with non-equal heights is investigated. Special purpose elements of distributed plasticity are employed for the study of the columns. Two distinct types of the problem are identified: Type A, where collisions may occur only between storey masses; and Type B, where the slabs of the first structure hit the columns of the other (72 Type A and 36 Type B pounding cases are examined). Type A cases yielded critical ductility requirements for the columns in the pounding area mainly for the cases where the structures were in contact from the beginning of the excitation. In both pounding types the ductility requirements of the columns of the taller building are substantially increased for the floors above the highest contact storey level probably due to a whiplash behaviour. The most important issue in the pounding type B is the local response of the column of the tall structure that suffers the hit of the upper floor slab of the adjacent shorter structure. In all the examined cases this column was in a critical condition due to shear action and in the cases where the structures were in contact from the beginning of the excitation, this column was also critical due to high ductility demands. It can be summarized that in situations of potential pounding, neglecting its possible effects leads to non-conservative building design or evaluation that may become critical in some cases. Copyright © 2004 John Wiley & Sons, Ltd.

Solving anti-plane problems of piezoelectric materials by the Trefftz finite element approach

Applications of the Trefftz finite element method to anti-plane electroelastic problems are presented in this paper. A dual variational functional is constructed and used to derive Trefftz finite element formulation. Special trial functions which satisfy boundary conditions are also used to develop a special purpose element with local defects. The performance of the proposed element model is assessed by an example and comparison is made with results obtained by other approaches. The Trefftz finite element approach is demonstrated to be ideally suited for the analysis of the anti-plane problem.

Steel-net rockfall protection - experimental and numerical simulation

The design of todays state-of-the-art rockfall barriers is currently based on experience and time consuming experimentation. Advanced simulation models are now under development to reduce prototyping time and development costs and will also provide a better understanding of complex mechanical systems under impact. In order to develop the models, extensive field testing was carried out. The individual components of the barriers were dynamically tested on a specially designed field apparatus, High energy full-scale tests on instrumented multi-span systems were additionally performed. The newly developed finite element program simulating the barriers is based on an explicit time step algorithm combined with special-purpose elements with large deformations and nonlinear material behavior for the netand brake-rings. For the interaction of rock and elements it also considers contact and sliding effects as well as friction.

Analysis of Tire Force and Moment Response During Side Slip Using an Efficient Finite Element Model

Abstract This paper describes the application of a new type of non-linear 3D finite element tire model for simulating tire spindle force and moment response during side slip. The simulation model, briefly described in the paper, is composed of shell elements, which model the tread deformation, coupled to special purpose finite elements that model the deformation of the sidewall and contact between the tread and the ground. The sidewall special purpose element uses a pre-computed look-up table to efficiently calculate the sidewall shape and the forces acting on the tread. The model is designed to predict the forces at the spindle and ground and the overall tire shape, as opposed to the internal stress fields. This paper considers the following deformation scenarios: 1) a vertically loaded tire deforming laterally on a flat surface and 2) a tire rolling straight ahead under a prescribed sideslip angle. Experimental data is also presented to verify the force predictions.

Trefftz Finite and Boundary Element Method

Part 1: finite element technique shape functions and element stiffness matrix brief historical background basic relationships in engineering problems modified variational principles the concept of T-complete solution comparison of T-elements with conventional finite elements comparison of T-elements with boundary elements. Part 2: potential problems - introduction statement of the problem T-complete functions assumed fields generation of element matrix equation rank condition special purpose functions sensitivity to mesh distortion orthotropic case the Helmholtz equation HT-element with boundary traction frame frameless T-elements. Part 3: linear elastostatics - introduction linear theory of elasticity assumed fields in plane elasticity T-complete functions variational formulations element stiffness equation special-purpose elements p-extension approach three-dimensional elasticity numerical examples. Part 4: thin plates - introduction thin plate theory assumed field T-complete functions and particular solutions variational formulations for plate bending generation of element stiffness matrix p-method elements special purpose functions Extension to thin plates on elastic foundation Two alternative plate bending p-elements Numerical examples and assessment. Chapter 5 - Thick Plates - Introduction Basic equations for Reissner-Mindlin plate theory Assumed fields and particular solution Variational formulation for HT thick plate elements Implementation of the new family of HT elements A 12 DOF quadrilateral element free of shear locking Extension to thick plates on elastic foundation Sensitivity to mesh distortion Numerical assessment. Chapter 6 - Transient Heat Conduction - Introduction Elements of heat conduction Time step formula Element matrix formulations T-complete functions and particular solutions Numerical examples. Chapter 7 - Geometrically Nonlinear Analysis of Plate Bending Problems - Introduction Basic equations of nonlinear thin plate bending Assumed fields and Trefftz functions Particular solutions Modified variational principle Element matrix Iterative scheme Extension to post-buckling thin plates on elastic foundation Geometrically nonlinear analysis of thick plates Numerical examples. Chapter 8 - Elastoplasticity - Introduction Time discretization Basic relations Assumed fields Constraints on the approximation functions Finite element equilibrium and compatibility equations Finite element equations Finite element governing system. Chapter 9 - Dynamics of Plate Bending Problems - Introduction Basic equations Time-stepping formulation Numerical examples. Chapter 10 - Trefftz Boundary Element Method - Introduction Potential problems Plane elasticity Thin plate bending Moderately thick plates.

Least-squares Trefftz-type elements for the Helmholtz equation

Trefftz-type elements, or T-elements, are finite elements the internal field of which fulfills the governing differential equations of the problem a priori whereas the prescribed boundary conditions and the interelement continuity must be enforced by some suitable method. In this paper, the relevant matching is achieved by means of a least-squares procedure. The so-called ‘frameless’ or least-squares T-elements for Helmholtz's equation (related to the scattering of waves by offshore structures) in 2-D are developed and studied. The required accuracy of the solution can be obtained by increasing the number of either the subdomains or T-functions, which can be regarded as the h- or p-type approach, respectively. Convergence studies are performed with much attention to the use of special purpose elements for a doubly connected domain with a circular hole and for an angular corner subdomain. The most attractive features of the presented formulation are its simplicity and robustness. The matrix of the resulting linear system is always Hermitian and positive definite. © 1998 John Wiley & Sons, Ltd.

A finite element musculoskeletal model of the shoulder mechanism

The finite element method described in this study provides an easy method to simulate the kinetics of multibody mechanisms. It is used in order to develop a musculoskeletal model of the shoulder mechanism. Each relevant morphological structure has been represented by an appropriate element. For the shoulder mechanism two special-purpose elements have been developed: a SURFACE element representing the scapulothoracic gliding plane and a CURVED-TRUSS element to represent muscles which are wrapped around bony contours. The model contains four bones, three joints, three extracapsular ligaments, the scapulothoracic gliding plane and 20 muscles and muscle parts. In the model, input variables are the positions of the shoulder girdle and humerus and the external load on the humerus. Output variables are muscle forces subject to an optimization procedure in which the mechanical stability of the glenohumeral joint is one of the constraints. Four different optimization criteria are compared. For 12 muscles, surface EMG is used to verify the model. Since the optimum muscle length and force-length relationship are unknown, and since maximal EMG amplitude is length dependent, verification is only possible in a qualitative sense. Nevertheless, it is concluded that a detailed model of the shoulder mechanism has been developed which provides good insight into the function of morphological structures.

Simulation of Microcrack Propagation in Creeping Polycrystals Due to Diffusive Grain Boundary Cavitation

Creep rupture in random polycrystalline aggregates is investigated numerically in terms of multi-grain cell studies using a Delaunay network modelling technique. The model involves a representation of the crystalline aggregate by means of special purpose elements attributed to each grain facet. These Delaunay elements account for elastic and creep deformations of the grains, free grain boundary sliding, as well as for the nucleation and diffusive growth of grain boundary cavities until coalescence leads to a facet microcrack. Damage accumulation is simulated numerically, until an excessive number of microcracks cause des-integration of the polycrystal. Primary attention is on the influence of randomness in the microstructure on creep rupture, either in terms of random variations of the size and shape of hexagonal grains, or in terms of random variations in the nucleation properties of grain boundaries. It is found that randomness always tends to decrease the life time. In particular, it is found that the life time depends sensitively on random variations of the geometry of the microstructure.

A special purpose element for shell-beam systems

This paper presents an isoparametric finite element for reinforced shells and plates. The formulation is based on general beam theory and takes into account both transverse shear deformation and torsional warping. The element exhibits complete two-line compatibility. Numerical examples are presented in order to demonstrate the validity of the formulation and the possibilities of application.

Singularity Finite Elements for Plate Bending

The elastic analysis of floor slabs using the finite element method encounters special difficulties at certain types of reentrant corners where classical plate theory predicts singular moments and shear forces. Examples are the corners of floor openings and the corner points of rectangular columns or shear cores. The nature of the stress singularities at these corners is discussed, and a family of special purpose hybrid stress plate bending elements is derived for use at these locations. These elements, which may be rectangular or L‐shaped, contain the appropriate singular moments as part of their assumed moment fields. The elements are tested on three example problems and the results are compared with those obtained using regular hybrid stress elements. Improved convergence and a better representation of the moment field are obtained with these special purpose elements. It is concluded that the more rapid convergence and additional accuracy justify the increase in computational effort.

Superconducting magnets for accelerators, beam lines and detectors

After twenty years of intensive R & D, superconducting magnets are now widely used in High Energy Physics laboratories. Large detectors and spectrometers have been the first area of successful application. New giant accelerators under construction or planned are definitely based on superconducting components whilst the use for beam lines appears more restricted to special purpose elements. A review of these various applications is presented.