A Hybrid Displacement VariationalFormulation Of BEM For Elastodynamics

Magneto-electro-elastic semi-analytical models for free vibration and transient dynamic responses of composite cylindrical shell structures

Free vibration analysis of two-dimensional functionally graded structures by a meshfree boundary–domain integral equation method

In this work, the dual boundary element method formulation is developed for effective modelling of dynamic crack problems. The static fundamental solutions are used and the domain integral, which comes from the inertial term, is transformed into boundary integrals using the dual reciprocity technique. Dynamic stress intensity factors are computed from crack opening displacements. Comparisons are made with quasi‐isotropic as well as anisotropic results, using the sub‐region technique. Several examples are presented to assess the accuracy and efficiency of the proposed method. Copyright © 2004 John Wiley & Sons, Ltd.

The dual reciprocity method (DRM) and the radial integration method (RIM) are techniques used in BEM to transform domain integrals into boundary integrals. They are suitable for boundary element formulations where a complete fundamental solution is either unavailable or very complex, because in these cases one or more terms can remain as domain integrals in order to use a simpler fundamental solution. Thus, a large number of problems can be solved with the knowledge of a few number of fundamental solutions and additional terms as inertia or non-linear effects, can be treated as body forces and taken to the boundary. In both methods, the remaining terms are approximated through a finite series expansion involving proposed approximating functions and coefficients to be determined. This expansion is substituted in the generated domain integrals that are, subsequently, tr ansformed into boundary integrals. In this paper, a boundary element formulation for thin shallow shells with no domain discretization is presented. The domain integrals due to the curvature of the shells are tran sformed into boundary integrals using the radial integration method. Two approximation functions are used in this work. The first is the radial ba sis function that has been used extensively in the DRM given by:

The free vibration of functionally graded porous cylindrical shell panels reinforced with graphene platelets (GPLs) was numerically investigated. The free vibration problem was formulated using the first-order shear deformation shell theory in the framework of the 2-D natural element method (NEM). The effective material properties of the GPL-reinforced shell panel were evaluated by employing the Halpin–Tsai model and the rule of mixtures and were modified by considering the porosity distribution. The cylindrical shell surface was transformed into the 2-D planar NEM grid to avoid complex computation, and the concept of the MITC3+shell element was employed to suppress shear locking. The numerical method was validated through benchmark experiments, and the free vibration characteristics of FG-GPLRC porous cylindrical shell panels were investigated. The numerical results are presented for four GPL distribution patterns (FG-U, FG-X, FG-O, and FG-Λ) and three porosity distributions (center- and outer-biased and uniform). The effects of GPL weight, porosity amount, length–thickness and length–radius ratios, and the aspect ratio of the shell panel and boundary condition on the free vibration characteristics are discussed in detail. It is found from the numerical results that the proposed numerical method accurately predicts the natural frequencies of FG-GPLRC porous cylindrical shell panels. Moreover, the free vibration of FG-GPLRC porous cylindrical shell panels is significantly influenced by the distribution pattern as well as the amount of GPLs and the porosity.

On the computation of mixed-mode stress intensity factors in functionally graded materials

Based on the Donnell’s approximations of the thin shell theory, this paper presents solutions for the problem of free non-axisymmetric vibration of stepped circular cylindrical shells with cracks. The shell under consideration is sub-divided into multiple segments separated by the locations of thickness variations. It is assumed that at thejth step there exists a circumferential surface crack with uniform depthcj. The influence of circular cracks with constant depth on the vibration of the shell is prescribed with the aid of a matrix of local flexibility. The latter is related to the coefficient of the stress intensity known in the linear fracture mechanics. Numerical results are obtained for cylindrical shells of stepped thickness containing cracks at re-entrant corners of steps. Shells with various combinations of boundary conditions can be analyzed by the proposed method. Furthermore, the influences of the shell thicknesses, locations of step-wise variations of the thickness and other parameters on the natural frequencies are examined. The results can be used for the approximate evaluation of dynamic parameters of cylindrical shells with cracks and flaws.

SUMMARYThe distance of a matrix to a nearby defective matrix is an important classical problem in numerical linear algebra, as it determines how sensitive or ill‐conditioned an eigenvalue decomposition of a matrix is. The concept has been discussed throughout the history of numerical linear algebra, and the problem of computing the nearest defective matrix first appeared in Wilkinsons famous book on the algebraic eigenvalue problem. In this paper, a new fast algorithm for the computation of the distance of a matrix to a nearby defective matrix is presented. The problem is formulated following Alam and Bora introduced in (2005) and reduces to finding when a parameter‐dependent matrix is singular subject to a constraint. The solution is achieved by an extension of the implicit determinant method introduced by Spence and Poulton in (2005). Numerical results for several examples illustrate the performance of the algorithm. Copyright © 2013 John Wiley & Sons, Ltd.

A dual reciprocity boundary element method based on global shapes functions belonging to a complete set to transform domain integrals into boundary ones is presented. Governing equations of Poisson and Helmholtz type are tested. It is demonstrated that the complete set shape functions are more efficient and accurate in approximating domain integrals than those based on incomplete set. INTRODUCTION The application of Boundary Element Method to linear partial differential operators, or to nonlinear and time-dependent ones, often leads to domain integrals that are cumbersome to handle. The dual reciprocity method pioneered by Nardini and Brebbia* is a powerful numerical technique which converts domain integrals into boundary ones using a set of global shape functions to approximate the integrand. It has been successfully applied in many problems involving dynamic, diffusion and nonlinear type operators. See Partridge, et al. for a comprehensive bibliographical review. The success of the dual reciprocity hinges on the ability to express the global shape functions into the Laplacian of a set of known functions such that the divergence theorem can be applied to resolve the domain integral into a boundary one. Such a constraint has so far limited the selection of shape functions to a set of radial functions of the form 1 + ar -f br^ -f • • •• Transactions on Modelling and Simulation vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X 344 Boundary Elements These radial functions do not form a complete set hence may lack convergence properties. Although practical evidences have indicated that those shape functions work quite satisfactorily, it is still desirable to use shape functions from a set that is complete in the mathematical sense. This paper present a dual reciprocity BEM based on global shape functions derived from a polynomial series. Perhaps due to its mathematical completeness, we are able to demonstrate that the numerical results are more efficient and accurate than the traditional radial set. DUAL RECIPROCITY For many non-standard, transient, or nonlinear partial differential equations it is possible to extract a standard, time-independent, linear partial differential operator C and lump the remainder to the right-hand-side: jC% = 6(z, ?/, Z, f , %, . . . , %,., %„ , %,, %,, %= . • •) (1) For the purpose of BEM implementation, these non-standard portions can be formally regarded as body forces. By the application of generalized Green's theorem = / Jr (2) where v is a conjugate variable, £* is the adjoint operator of £, B» and B* are operators resulting from integration by parts, which may be viewed as generalized normal derivatives, F is the solution boundary, and fi the solution domain, a boundary integral equation can be formed as: cu = / (u*q uq*) j bu* dSl (3) where quantities denoted by asterisk are free-space Green's functions satisfying £*u* = —6, with 6 a Dirac delta function, c is a constant dependent on the Cauchy principal value integration of the singularity at the base point, q and q* are generalized normal derivatives of u and u*. To circumvent the domain integration, the function b can be approximated by an expansion based on a set of global shape functions /, such that &*fX£ (4) j=l in which a, are coefficients determined by collocation, and n^ is the number of collocation nodes. The shape functions are chosen such that particular solutions of the set of inhomogeneous equations Ciij = /,• (5) Transactions on Modelling and Simulation vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X Boundary Elements 345 are explicitly known. The domain integral in (3) can be transformed as [ / j^bu da = i ŷ /ju (6) j=l L ^1 J The last expression was a consequence of the application of (2). Substitution of the above into (3) yields / (u*q uq} dY = Y a, \cu, I (%*& titf) dT (7) •/r fr(. L 7r J cu We hence in principal have achieved the elimination of the domain integral. However, the detail of the implementation is somewhat dependent on the type of body force terms. We shall illustrate below the implementation by examples. EXAMPLES Operator of V^u = b(x, y) type For the governing equation of Poisson type, q in (7) is du/dn, and u* is — lnr/2%for 2-D or l/4%r for 3-D geometries. For simplicity, only two-dimensional problems will be studied herein. If the right-hand-side 6(x, y) is a mathematical expression whose particular solution is known as Up, the solution of (8) can be expressed as % = %& + %? (9) where u^ is the homogeneous solution and V2%& = 0 (10) It is apparent that Uh = u — Up satisfies the boundary integral equation c(u «„) = / [u(q q,} -(uu,)q'] (11) If we approximate the right-hand-side b(x,y) as (4), and find the particular solution term by term as V = /,(12) Transactions on Modelling and Simulation vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X 346 Boundary Elements it is clear that j=l By the above interpretation, we have provided an alternative way for deriving (7), which is rewritten below with explicit reference to coordinates as c(x)u(x)f(u*(x',x)q(x')-u(x')q-(x',x)] dT(x') j r

A meshless BEM for solving transient non-homogeneous convection-diffusion problem with variable velocity and source term

Static and dynamic characteristics of the post-buckling of fluid-conveying porous functionally graded pipes with geometric imperfections

One of the important problems in linear algebra is the determination of eigenvalues and eigenvectors of a matrix. Eigenvalue problems in differential and integral equations can also be approximated by algebraic eigenvalue problems. In spite of the simplicity of its formulation, many algorithms are required to deal efficiently with a wide spectrum of problems, which are encountered in practice. The variety in problems arises because of the variety in type of matrix and the varying requirements from the solution. The classification of matrices into different types has been described in Section 3.1 and will not be repeated here. The varying requirements from the solution arise because we may need all eigenvalues and eigenvectors or only some of them. Alternately, we may be interested in only eigenvalues. In some problems, we may be interested in only the eigenvalue with the largest magnitude, or the one with the largest real part, while in other problems we may require all eigenvalues in a given region of the complex plane. In general, there is some correlation between the type of matrix and the amount of information required from the solution. For example, with large matrices which usually arise from discretisation of differential or integral equations, we are normally interested in only a few eigenvalues and possibly the eigenvectors. On the other hand, for small matrices we normally want all eigenvalues and some or all eigenvectors.

Free vibration and stability of angle-ply laminated composite and sandwich plates under thermal loading

Based upon a variational principle and the associated theory derived in three preceding papers, an expression for the magneto-elastic buckling value for a system of an arbitrary number of parallel superconducting beams is given. The total current is supposed to be equal both in magnitude and direction for all beams, and the cross-sections are circular. The expression for the buckling value is formulated more explicitly in terms of the so-called buckling amplitudes, the latter following from an algebraic eigenvalue problem. The pertinent matrix is formulated in terms of complex functions, which are replaced by real potentials. The matrix elements are calculated by a numerical method, solving a set of intergral equations with regular kernels. Apart from the buckling value(s) the buckling modes are also obtained. Finally, our results are compared with the results of a mathematically less complicated theory, i.e. the method of Biot and Savart.

A new method of perception of Tissot's Theory of Distortions and the computation of the parameters of Tissot's lndicatrix, used to analyze map distortions of cartographic projections, is proposed. The new approach is based, on the algebraic eigenvalue problem and makes use of the Singular Value Decomposition of the column-scaled Jacobian matrix of the mapping equations. The semiaxes of Tissot's lndicatrix are evaluated directly, and the use of quadratic forms, which may cause unnecessary loss of numerical precision, is avoided. Several advantages of the new approach over the original Tissot's approach are detailed. The traditional concept of Tissot's lndicatrix also is examined for comparison. It is presented from the non-traditional point of view, as the study of a variation of the positive definite quadratic form under a mapping transformation. The complete collection of the computational formulas of the original Tissot's approach (improved for numerical efficiency) is included.