Solving Nonlinear Differential Equations of Mechanics With the Boundary Element Method and Radial Basis Functions

Pollandt, R.

doi:10.1002/(sici)1097-0207(19970115)40:1<61::aid-nme50>3.0.co;2-q

Cited by 24 publications

(6 citation statements)

References 14 publications

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“…where h is given by (2.2) and k ≥ σ > n/2 + 1. We now outline the third method, that is, a boundary element (BEM) method, following Pollandt (1997). The dual reciprocity method uses the second Green's formula and a fundamental solution φ( • ) of the Laplace operator ∆, in order to reformulate a boundary value problem as a boundary integral problem over a space of one dimension lower.…”

Section: Numerical Solution Of Partial Differential Equationsmentioning

confidence: 99%

“…Some possible applications, for instance, for numerical solutions of partial differential equations come to mind when those radial functions are used because they can act as finite elements, and they are being tested at the moment by several colleagues for this very purpose. Results are mildly encouraging, although the mathematical analysis is still lacking (see, e.g., Fasshauer (1999)); some mathematical underpinning is given in Franke and Schaback (1998) and Pollandt (1997). We will explain further and give examples in Section 6.…”

Section: Introductionmentioning

confidence: 99%

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Radial basis functions

2000

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Radial basis function methods are modern ways to approximate multivariate functions, especially in the absence of grid data. They have been known, tested and analysed for several years now and many positive properties have been identified. This paper gives a selective but up-to-date survey of several recent developments that explains their usefulness from the theoretical point of view and contributes useful new classes of radial basis function. We consider particularly the new results on convergence rates of interpolation with radial basis functions, as well as some of the various achievements on approximation on spheres, and the efficient numerical computation of interpolants for very large sets of data. Several examples of useful applications are stated at the end of the paper.

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Section: Numerical Solution Of Partial Differential Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Radial basis functions

2000

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“…The FE method coupled with Newton's iterations has been extensively used to solve geometrically and materially nonlinear problems (Hughes and Liu, 1981;Reddy and Singh, 1981;Noor and Peters, 1980). Pollandt (1997) has solved von Karman plate equations using a boundary element method constructed via radial basis functions. In another development, Gao (2000) has developed a general duality principle to search for the extremas (along with their stability types) in non-convex energy expressions, used for post-buckling analysis of thick beams.…”

Section: Introductionmentioning

confidence: 99%

Multi-step transversal and tangential linearization methods applied to a class of nonlinear beam equations

Viswanath

Roy

2007

International Journal of Solids and Structures

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A novel and continuously parameterized form of multi-step transversal linearization (MTrL) method is developed and numerically explored for solving nonlinear ordinary differential equations governing a class of boundary value problems (BVPs) of relevance in structural mechanics. A similar family of multi-step tangential linearization (MTnL) methods is also developed and applied to such BVP-s. Within the framework of MTrL and MTnL, a BVP is treated as a constrained dynamical system, i.e. a constrained initial value problem (IVP). While the MTrL requires the linearized solution manifold to transversally intersect the nonlinear solution manifold at a chosen set of points across the axis of the independent variable, the essential difference of the present MTrL method from its previous version [Roy, D., Kumar, R., 2005. A multistep transversal linearization (MTL) method in nonlinear structural dynamics. J. Sound Vib. 17, 829-852.] is that it has the flexibility of treating nonlinear damping and stiffness terms as time-variant damping and stiffness terms in the linearized system. The resulting time-variant linearized system is then solved using Magnus' characterization [Magnus, W., 1954. On the exponential solution of differential equations for a linear operator. Commun. Pure Appl. Math., 7, 649-673.]. Towards numerical illustrations, response of a tip loaded cantilever beam (Elastica) is first obtained. Next, the response of a simply supported nonlinear Timoshenko beam is obtained using a variationally correct (VC) model for the beam [Marur, S., Prathap, G., 2005. Nonlinear beam vibration problems and simplification in finite element model. Comput. Mech. 35(5), 352-360.]. The new model does not involve any simplifications commonly employed in the finite element formulations in order to ease the computation of nonlinear stiffness terms from nonlinear strain energy terms. A comparison of results through MTrL and MTnL techniques consistently indicate a superior quality of approximations via the transversal linearization technique. While the usage of tangential system matrices is common in nonlinear finite element practices, it is demonstrated that the transversal version of linearization offers an easier and more general implementation, requires no computations of directional derivatives and leads to a consistently higher level of numerical accuracy. It is also observed that higher order versions of MTrL/MTnL with Lagrangian interpolations may not work satisfactorily and hence spline interpolations are suggested to overcome this problem.

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“…One can thus obtain the matrix of particular solutions without inverting a dense matrix, thus saving considerable computational labor. Quasi-interpolation [48] is another technique by which one can construct the interpolant without the matrix inversion procedure.…”

Section: Discussionmentioning

confidence: 99%