Showing papers in "Communications in Numerical Methods in Engineering in 1997"

On contact between three-dimensional beams undergoing large deflections

Abstract: Contact between three-dimensional beams which undergo large motions is considered. To formulate the associated constraint conditions the point of contact has to be detected within the beam. Once this is known the contact constraint has to be formulated for a given beam discretization and the associated contribution to the weak form has to be developed. Also, consistent linearization of the contact contribution is derived, which is needed within Newton's method. © 1997 John Wiley & Sons, Ltd.

Completed Richardson extrapolation in space and time

Abstract: The technique of Richardson extrapolation, which has previously been used on time-independent problems, is extended so that it can also be used on time-dependent problems. The technique presented is completed in the sense that the extrapolated solution is calculated at all spatial grid nodes which coincide with nodes of the finest grid considered. Numerical examples are presented when the technique is applied to the Lax–Wendroff and Crank–Nicholson finite difference schemes which are used to approximate solutions to the convection–diffusion equation. The examples show that extrapolation can be an easy and efficient way in which to produce accurate numerical solutions to time-dependent problems. © 1997 John Wiley & Sons, Ltd.

Fourier expansion‐based differential quadrature and its application to Helmholtz eigenvalue problems

Abstract: SUMMARY Based on the same concept as generalized diAerential quadrature (GDQ), the method of Fourier expansionbased diAerential quadrature (FDQ) was developed and then applied to solve the Helmholtz eigenvalue problems with periodic and non-periodic boundary conditions. In FDQ, the solution of a partial diAerential equation is approximated by a Fourier series expansion. The details of the FDQ method and its implementation to sample problems are shown in this paper. It was found that the FDQ results are very accurate for the Helmholtz eigenvalue problems even though very few grid points are used. #1997 John Wiley & Sons, Ltd.

Homogenization of periodic masonry: plane stress, generalized plane strain or 3d modelling?

Abstract: Through the homogenization theory for periodic media, the macroscopic behaviour of masonry may be derived from the behaviour of its constitutive materials (brick and mortar). Such a procedure has been used by many authors but always in an approximate manner. In particular, masonry has been considered either as infinitely thin (two-dimensional media under plane stress), or as infinitely thick (two-dimensional media under generalized plane strain). In order to determine the range of validity of either assumption, the homogenization theory is here implemented in a rigorous way, i.e. taking into account the finite thickness of masonry. Both brick and mortar being assumed as subjected to isotropic damage, numerical computations show that the above-mentioned assumptions have little influence on the macroscopic elastic behaviour of masonry, but may significantly affect its non-linear response (ultimate load and mode of failure). © 1997 John Wiley & Sons, Ltd.

A four-node differential quadrature method for straight-sided quadrilateral Reissner/Mindlin plates

Abstract: A four-node differential quadrature (4NDQ) method is proposed as a simple, accurate and efficient numerical technique for bending analysis of Reissner/Mindlin plates in an arbitrarily straight-sided quadrilateral domain. For demonstration, a clamped skew rhombic plate is used as an example to illustrate the convergence, accuracy and efficiency of the 4NDQ method. © 1997 John Wiley & Sons, Ltd.

Simulations of the EW undular bore

Abstract: The equal width equation is solved by a Petrov–Galerkin method using quadratic B-spline spatial finite elements. A linear recurrence relationship for the numerical solution of the resulting system of ordinary differential equations is obtained via a Crank–Nicolson approach involving a product approximation. The motion of solitary waves is studied to assess the properties of the algorithm. The development of an EW undular bore is investigated and compared with that of the RLW bore. © 1997 John Wiley & Sons, Ltd.

Partitioned solution procedure for simultaneous integration of coupled-field problems

Abstract: SUMMARYAn implicit unconditionally stable partitioned solution procedure for the simultaneous integration oftransient coupled field problems is presented. The procedure does not require that the full system of coupledequations be assembled, and allows use of existing single-field analysis software modules to solve thecoupled fieldproblem. An iterative partitioned conjugategradient procedureis usedtoavoid having toformand assemble the Schur complement matrix. The coupling matrices never need be formed, thus resulting insubstantial computational savings. # 1997 by John Wiley & Sons, Ltd. Commun. numer. meth. engng.,vol. 13, 239–247 (1997). (No. of Figures: 2 No. of Tables: 1 No. of References: 20) 1. INTRODUCTIONThe transient response analysis of coupled-field problems is becoming a major challenge inmany engineering disciplines. In coupled problems two or more physical entities interact witheach other, with the independent solution of any one entity being impossible without simul-taneous solution of the others. Some problems, such as dynamic fluid-structure interaction,may be characterized by distinct spatial subdomains (e.g. fluid and structure) with distinctsets of dependent variables (e.g. fluid pressure and solid displacements). In that case, neithersubsystem can be solved independently of the other due to the unknown interface inter-action forces. Other problems are characterized by the interaction of essentially di•erentphysical phenomena within the same material domain, e.g. thermal and mechanical interaction.Although it is believed that the proposed procedure could also be useful for the first class ofproblems, it is this last class of problems which is of primary concern in this paper. In thisclass of problems the coupling occurs through the di•erential governing equations describingthe di•erent physical phenomena. Each component field is then spatially discretized into afinite number of degrees of freedom through a finite element method and called a subsystem.The goal is then to integrate in time the resulting coupled semi-discrete matrix equations ofmotion.A typical system of equations which arises in linear thermo-elasticity is given asM000uy⁄C0y

Unified finite elements based on the classical and shear deformation theories of beams and axisymmetric circular plates

Abstract: In this paper a unified finite element model that contains the Euler-Bernoulli, Timoshenko and simplified Reddy third-order beam theories as special cases is presented. The element has only four degrees of freedom, namely deflection and rotation at each of its two nodes. Depending on the choice of the element type, the general stiffness matrix can be specialized to any of the three theories by merely assigning proper values to parameters introduced in the development. The element does not experience shear locking, and gives exact generalized nodal displacements for Euler-Bernoulli and Timoshenko beam theories when the beam is homogeneous and has constant geometric properties. While the Timoshenko beam theory requires a shear correction factor, the third-order beam theory does not require specification of a shear correction factor. An extension of the work to axisymmetric bending of circular plates is also presented. A stiffness matrix based on the exact analytical form of the solution of the first-order theory of circular plates is derived.

Free vibration analysis of non‐cylindrical coil springs by combined use of the transfer matrix and the complementary functions methods

Abstract: The transfer matrix method and the complementary functions method have been employed in the free vibration analysis of helical springs of irregular shape. The rotary inertia effects, the axial and shear deformations and cross-sections of different shapes having double symmetry have been included in the analysis. The formulation has yielded the exact free vibration frequencies of non-cylindrical coil springs. © 1997 John Wiley & Sons, Ltd.

Two-grid finite element formulations of the incompressible Navier-Stokes equations

Abstract: We compare two different two-grid finite element formulations applied to the Navier-Stokes equations, namely a multigrid and a mixed or composite formulation. In the latter case the pressure is interpolated on a coarser grid than the velocity, using mixed elements instead of mixed interpolation functions. In the multigrid formulation an equal-order interpolation is used on the same elements. A pressure Poisson formulation is used in both cases. Computational results are presented to compare the methods and comparisons are also made with other results.

Homogenization of multicomponent composite orthotropic materials using fea

Abstract: This paper demonstrates a simple finite element implementation of Lagrange multipliers to model the mechanical behaviour of an orthotropic composite material. The research shows the proper set of kinematic boundary conditions that must be applied in 2D plane stress elasticity to achieve the correct unit strain vectors that, upon interrogation of the associated Lagrange multipliers, give the stresses induced by these strain vectors. From these stresses the terms in the elasticity matrix can be evaluated. As well as demonstrating the correct kinematic conditions required, the paper presents the consequences of applying intuitive but incorrect conditions. © 1997 John Wiley & Sons, Ltd.

An automatic tetrahedral mesh generation scheme by the advancing front method

Abstract: SUMMARY The paper deals with the discretization of any given multi-connected volume into a set of tetrahedral elements. A simple but robust tetrahedrization scheme based on a two-stage advancing front technique is presented. The method evolves from the triangulated domain bounding surfaces for which geometry representations are derived from triangular Bezier patches. Tetrahedral elements are then generated which fill the domain volume based on the set of distributed interior nodes. A new and eAcient procedure is introduced for the distribution of the mesh interior nodes which uses an inverse-power interpolation technique. The proposed scheme is robust in that it is capable of tetrahedrizing a given arbitrary domain of any degree of irregularity, and allows the distribution of its interior nodes to be specified by the user. Results are presented typical of those which might be encountered in hydrodynamics modelling involving flows with a free surface.

Hybrid approximation functions in the dual reciprocity boundary element method

Abstract: The dual reciprocity boundary element method traditionally uses the linear radial basis function r for interpolation Recently, however, the use of the r function has been questioned both in relation to accuracy and in relation to the number and position of internal nodes required to obtain satisfactory solutions Much research has been done in an attempt to fix criteria for choosing which approximation function should be used One of the alternatives recently suggested is the augmented thin plate spline function, which consists of a thin plate spline function, r2 log(r), augmented with the first three terms of a Pascal triangle expansion In this paper families of similar functions are obtained by augmenting radial basis functions with appropriate global expansions: these functions will be called hybrid approximate functions It will be shown that using an appropriate hybrid function accurate results can be obtained for many body forces or pseudo body forces in elasticity without the need for internal nodes© 1997 John Wiley & Sons, Ltd

Non-iterative solution of inverse heat conduction problems in one dimension

Abstract: A model is presented for the inverse determination of the strength of a temporal–spatial-dependent heat source in the one-dimensional heat conduction problem. This model is constructed from the finite difference approximation of the differential heat conduction equation based on the assumption that the temperature measurements are available over the problem domain. In contrast to the traditional approach, the iteration in the proposed model can be done only once and the inverse problem can be solved in a linear domain. In the examples, comparisons between the exact heat sources and the estimated ones (without measurement errors) are made to confirm the validity of the proposed model. The close agreement between the exact solutions and the estimated results shows the potential of the proposed model in finding an accurate value of the heat source in the one-dimensional heat conduction problem. © 1997 John Wiley & Sons, Ltd.

Improving modularity in object-oriented finite element programming

Abstract: Making a finite element code easier to maintain is achieved by further modularizing it. Due to its two levels of modularity (of procedures and data), object-oriented programming is the method of choice. Its potential is investigated in two bottlenecks of finite element programming where so far it has not proved significantly more successful than classical Fortran programming. The first of these is the lack of adequate data structures between the analysis specifications and the basic objects of the finite element method, like the element and the node. This is solved by defining two classes of objects, namely problem and domain, with clearly differentiated specifications. The second gap deals with solving linear equation systems. Introducing an intermediate class UnassembledMatrix allows alternative storage/solving schemes to be implemented in a very flexible manner. © 1997 by John Wiley & Sons, Ltd.

Higher-order compact mixed methods

Abstract: We develop a class of higher-order mixed finite difference methods for elliptic partial differential equations. The problem is recast as a first-order mixed system and the higher-order compact schemes follow as a natural extension of the formulations we developed previously for the scalar PDE problem. Since the flux appears explicitly in the mixed formulation we obtain higher-order (nodal superconvergent) solutions in both the primary solution field and also the flux. Some supporting numerical experiments are included to demonstrate the superconvergent rates. © 1997 John Wiley & Sons, Ltd.

1D infinite element for dynamic problems in saturated porous media

Abstract: A fully coupled 1D infinite element for frequency domain analysis of wave propagation problems in unbounded saturated porous media is presented. The element wave propagation function is derived using an analytical solution for Biot's formulation (1962). The effectiveness and the accuracy of the infinite element proposed are demonstrated through a simple wave propagation problem in a semi-infinite soil column subjected to a harmonic surface loading. It is shown that an accurate representation of the problem can be obtained by coupling the conventional finite elements with the proposed infinite element. The accuracy of the solution significantly deteriorates when free or fixed boundary conditions are imposed at the truncated boundary instead of the infinite element. © 1997 John Wiley & Sons, Ltd.

Optimization of mechanical systems: on non-linear first-order approximation with an additive convex term

Abstract: This paper presents an improved approximation technique for gradient based approximation methods of mathematical programming. The proposed technique prevents oscillations of the sequence of approximate solutions in the optimization process efficiently and preserves the relatively simple form of the approximating functions. The improvement is achieved by adding an appropriate convex term to each conventional approximating function. The theory is illustrated with several numerical examples.

Simulations of passively enhanced conjugate heat transfer across an array of volumetric heat sources

Abstract: Numerical simulations were performed to investigate convective–conductive heat transfer due to a laminar boundary layer flow of air over a two dimensional array of rectangular chip blocks which represent the finite heat sources. The main focus of this study is on the simulation of the flow fields and temperature variations of the air and the chip blocks. The purpose of this study is to verify the effects of the openings of the board in the areas between the chip blocks on the enhancement of cooling the heating blocks. Due to a pressure differential occurring across the opening, the induced vertical flow serves as a suction or blowing force and consequently enhances heat dissipation to the ambient fluid. The optimal configuration of the chip board regarding cooling the heat source would yield lower chip temperatures with limited chip-to-chip temperature variations. A time-accurate numerical scheme algorithm, PISO (pressure-implicit with splitting of operators), is used to simulate the conjugate heat transfer between the fluid and solid phases. In this work, a set of false solid properties was employed to force the solid side to have a time scale comparable to that of the fluid side in order to avoid numerical instabilities due to different time scales used in the calculations. The results of the simulations show that the existence of the array of blocks results in stagnant flow regions between blocks in which heat convected to the ambient flow field is limited. It was found that heat transfer can be enhanced passively, especially in the areas between blocks, by opening the chip board between blocks. The enhancement of heat transfer thus occurring is presumably due to a pseudo-suction force which induces a vertical flow between blocks. The enhancement of heat transfer for the chips on-board is reflected by a global increase of the Nusselt number on the chip blocks, especially on the west sides of the chips located further downstream of the flow direction. Further investigation shows that the chip-to-chip temperature variations diminish if the openings located upstream of the front end block and downstream of the rear end block are sealed. The optimal cooling configuration for the array of chip blocks can be utilized by the electronics industry. © 1997 John Wiley & Sons, Ltd.

Interpolation-free space-time remeshing for the Burgers equation

Abstract: We present a mesh management strategy for use with discontinuous-Galerkin space-time finite element formulations of flow problems. We propose a refinement technique for simplex-type meshes which requires no interpolation between grids at slab interfaces. The strategy uses an a posteriori spatial error estimator to tag refinement or coarsening. Orientation of element edges along flow characteristics is accomplished by nodal displacement, and by a new diagonal-swapping technique to correct for the effects of misalignment due to h-refinement. The swapping procedure realigns the mesh to improve the effectiveness of the h-adaptive process. Results are presented for the Burgers Equation using large time steps on a problem which exhibits merging and steepening fronts

Third-order time-step integration methods with controllable numerical dissipation

Abstract: In this paper, the second-order-accurate non-dissipative Newmark method is modified to third-order-accurate with controllable dissipation by using complex time steps. Among these algorithms, the asymptotic annihilating algorithm and the non-dissipative algorithm are found to be the first sub-diagonal (1,2) and diagonal (2,2) Pade approximations, respectively. The non-dissipative algorithm is therefore fourth-order-accurate. The stability properties and errors for algorithms with other dissipations are between these two algorithms. The spectral radii, the algorithmic damping ratios and the relative period errors for the present third-order complex-time-step algorithms are compared favourably with other algorithms.

Structural responses to non-uniformly modulated evolutionary random seismic excitations

Abstract: This paper presents an accurate algorithm for structural non-stationary random seismic responses. The ground acceleration is assumed to be an evolutionary random process, of which the modulation function is non-uniformly modulated, i.e. it depends on both time and frequency. The method is used to compute the non-stationary seismic responses of a non-uniform shear beam. The dependency of the power spectral densities of the excitations and of the responses on time and frequency is discussed, based on numerical results.

Modelling of steam surface condenser using finite element methods

Abstract: Two numerical procedures are proposed for surface condenser analysis. In the first method a two-noded element is used with a variable heat transfer coefficient for each element. This method is suitable for performance analysis under various operating conditions. A four-noded element is also proposed for studying the axial heat conduction effect in tubes. In the second method, shell side flow and heat transfer are simulated in order to study the flow distribution and pressure drop characteristics. The governing equations are solved in primitive variable form. The finite element method with a Eulerian velocity correction algorithm is used. These two procedures are applied to two different types of surface condensers and the results are found to be in good agreement with experimental data. © 1997 John Wiley & Sons, Ltd.

The united point force solution for both isotropic and transversely isotropic media

Abstract: This paper treats a united-form solution for a point force applied at the interior of an infinite transversely isotropic solid. Several heuristic functions are adopted to obtain the expressions of the solution based on the general solution. To exclude some indeterminate attributes, the expressions are rewritten. In the final expressions, unlike previous publications where the solutions are expressed in different forms, or when some individual constants have different definitions depending on the conditions satisfied by the elastic constants, we provide united solutions which are suitable for all stable transversely isotropic materials and isotropic materials. Thus accurate numerical evaluation of the boundary element method can be performed directly without the need to resolve the singularity algebraically. Some numerical examples with BEM are also presented in this paper. © 1997 John Wiley & Sons, Ltd.

An exact structural static reanalysis method

Abstract: This paper presents an exact structural static reanalysis method for locally modified structures. Through the introduction of structural rigid body motion eigenvectors, the generalized structural compliance matrix can be obtained and the original stiffness equation is transformed into a linear system of much lower order. The general solution of displacements can be expressed prior to any assignment of boundary conditions. For a structure with given boundary and loading conditions, the displacements can be obtained by solving this linear system. For locally modified structures, the structural compliance matrix can be adjusted quickly. This static reanalysis method can be used for structures with modifications on structural elements, boundary and loading conditions, either independently or in combination. Two test examples are provided in the paper to prove the efficiency of the method.

Padé approximants applied to a non‐linear finite element solution strategy

Abstract: The present work deals with an asymptotic numerical method, based on Pade approximants. The expected advantage of this method is twofold. Firstly, it reduces the computational costs. Secondly, the automatization of the continuation process becomes easier, since the step-length can be determined a posteriori. So far, this method has only been applied to DKT elements. Here it is applied to other types of elements, namely truss elements and finite rotation non-linear shell elements. It will be shown that difficulties arise when this method is applied to finite rotation shell elements.

Solution of torsion crack problem of an orthotropic rectangular bar by using computing compliance method

Abstract: For a cracked orthotropic torsion bar, the torsion compliance C is evaluated numerically in this paper. The dependence of the compliance C with respect to the crack length A will give the stress intensity factor at the crack tip. Numerical examples are given to demonstrate the influence of the elastic material constants of orthotropic materials. ©1997 John Wiley & Sons, Ltd.

A numerical algorithm for double surface integrals over quadrilaterals with a 1/R singularity

Abstract: A new algorithm for the double surface integral with a 1/R singularity over flat or curved quadrilateral elements is presented. This algorithm is of special interest in the numerical implementation of the variational boundary element methods for acoustic radiation of finite-volume bodies. The presented algorithm takes the factor of singularity in an integrand as the weight function, and approximates the integral by the area of the surface element times a weighted average of the remaining integrand at a number of the integral points in the element domain. A numerical example demonstrates the accuracy of the proposed algorithm. © 1997 John Wiley & Sons, Ltd.

Special Trefftz Elements and Improvement of Their Conditioning

Abstract: This paper proposes a new family of special Trefftz elements in which the boundary conditions on an internal curve (stiffener, hole, etc.) are fulfilled by the least squares procedure. The author anticipates difficulties with conditioning of such elements and therefore proposes a method to improve the aspect. © 1997 John Wiley & Sons, Ltd.

Sequential function approximation for the solution of differential equations

Abstract: A computational method for the solution of differential equations is proposed. With this method an accurate approximation is built by incremental additions of optimal local basis functions. The parallel direct search software package (PDS), that supports parallel objective function evaluations, is used to solve the associated optimization problem efficiently. The advantage of the method is that, although it resembles adaptive methods in computational mechanics, an a priori grid is not necessary. Moreover, the traditional matrix construction and evaluations are avoided. Computational cost is reduced while efficiency is enhanced by the low-dimensional parallel-executed optimization and parallel function evaluations. In addition, the method should be applicable to a broad class of interpolation functions