an element free galerkin formulation for stable crack growth in an elastic solid abstract the discrete formulation for stable crack growth in an elastic solid using the element free galerkin efg method based on the moving least squares approximations has been established the efg method provides the ability to successfully model steep gradients such as those that exist at a crack tip through the introduction of an additional distribution of nodal points in this formulation the inertia force term in the momentum equation is converted into a spatial gradient term by employing the steady state conditions a convective domain is employed to account for the analysis domain moving at the same speed as the crack front a number of stable crack growth problems are examined and comparisons between the numerical predictions and analytical solutions are made for the near tip fields the crack opening profiles as well as the global control parameters such as stress intensity factor k and energy release rate g

singular perturbations for third order nonlinear multi point boundary value problem abstract this paper is devoted to study the following third order multi point singularly perturbed boundary value problem x t f t x t x t x t 0 0 t 1 0 1 x 0 0 ax 0 bx 0 i 1 n 2 i x i a cx 1 dx 1 i 1 n 2 i x i b where a b c d 0 a b r a b 0 c d 0 i 0 i 0 i 1 2 n 2 0 1 2 n 2 1 and 0 1 2 n 2 1 the existence uniqueness and asymptotic estimates of solutions of the boundary value problem are give by using priori estimates differential inequalities technique and leray schauder degree theory

a boundary value technique for singular perturbation problems abstract a new way to solve singular perturbation problems is introduced it is designed for the practicing engineer or applied mathematician who needs a practical tool for these problems easy to use modest problem preparation and ready computer implementation as with other methods the original problem is partitioned into inner and outer solution differential equation systems however asymptotic solution techniques are not employed the method is distinguished by the following facts the inner solution problem is solved as a two point boundary problem where the terminal boundary conditions are supplied by the solution of the outer solution problem in turn the outer solution differential equations are solved as an initial value system where the initial conditions are supplied by the inner solution at its terminal point the method is iterative on the terminal point of the inner solution problem three numerical examples are included

superlinear convergence of primal dual interior point algorithms for nonlinear programming the local convergence properties of a class of primal dual interior point methods are analyzed these methods are designed to minimize a nonlinear nonconvex objective function subject to linear equality constraints and general inequalities they involve an inner iteration in which the log barrier merit function is approximately minimized subject to satisfying the linear equality constraints and an outer iteration that specifies both the decrease in the barrier parameter and the level of accuracy for the inner minimization under nondegeneracy assumptions it is shown that asymptotically for each value of the barrier parameter solving a single primal dual linear system is enough to produce an iterate that already matches the barrier subproblem accuracy requirements the asymptotic rate of convergence of the resulting algorithm is q superlinear and may be chosen arbitrarily close to quadratic furthermore this rate applies componentwise these results hold in particular for the method described in a r conn n i m gould d orban and p l toint math program ser b 87 2000 pp 215 249 and indicate that the details of its inner minimization are irrelevant in the asymptotics except for its accuracy requirements

graded mesh difference schemes for singularly perturbed two point boundary value problems etude de l approximation numerique des operateurs differentiels sans points de retournement par des schemas aux differences finies compacts analyse de la stabilite

properties of the log barrier function on degenerate nonlinear programs we examine the sequence of local minimizers of the log barrier function for a nonlinear program near a solution at which second order sufficient conditions and the mangasarian fromovitz constraint qualification are satisfied but the active constraint gradients are not necessarily linearly independent when a strict complementarity condition is satisfied we show uniqueness of the local minimizer of the barrier function in the vicinity of the nonlinear program solution and we obtain a semiexplicit characterization of this point when strict complementarity does not hold we obtain several other interesting characterizations in particular an estimate of the distance between the minimizers of the barrier function and the nonlinear program in terms of the barrier parameter and a result about the direction of approach of the sequence of minimizers of the barrier function to the nonlinear programming solution

an augmented lagrangian element free alef approach for crack discontinuities a new meshless formulation for numerical analysis in fracture mechanics is presented the formulation is based on the construction of shape functions by a moving least squares approximation mlsa the peculiarity of the present approach lies in the definition of an augmented lagrangian total potential energy functional with this technique called augmented lagrangian element free alef some drawbacks on imposing boundary and interfaces conditions are worked out a large number of lagrangian multipliers can be introduced without increasing the number of primary variables and inducing the strict convexity of the total potential energy functional a new model for simulating fracture discontinuity based on the imposition of interface conditions on the edges of a virtual crack is defined which permits to avoid some crack simulation difficulties inherent in the element free galerkin method some numerical examples are reported to evaluate the accuracy of the formulation

approximate method for the numerical solution of singular perturbation problems we present an approximate method for the numerical solution of linear singularly perturbed two point boundary value problems in ordinary differential equations with a boundary layer on the left end of the underlying interval it is motivated by the asymptotic behavior of singular perturbation problems the original problem is divided into inner and outer region problems the reduced problem is solved to obtain the terminal boundary condition then a new inner region problem is created and solved as a two point boundary value problem in turn the outer region problem is also modified and the resulting problem is efficiently treated by employing the trapezoidal formula coupled with discrete invariant imbedding algorithm the proposed method is iterative on the terminal point some numerical experiments have been included to demonstrate its applicability

element free galerkin method for wave propagation and dynamic fracture abstract element free galerkin method efg is extended to dynamic problems efg method which is based on moving least square interpolants mls requires only nodal data no element connectivity is needed this makes the method particularly attractive for moving dynamic crack problems since remeshing can be avoided in contrast to the earlier formulation for static problems by authors the weak form of kinematic boundary conditions for dynamic problems is introduced in the implementation to enforce the kinematic boundary conditions with this formulation the stiffness matrix is symmetric and positive semi definite and hence the consistency conergence and stability analyses of time integration remain the same as those in finite element method numerical examples are presented to illustrate the performance of this method the relationship between the element free galerkin method and the smooth particle hydrodynamics sph method is also discussed in this paper results are presented for some one dimensional problems and two dimensional problems with static and moving cracks

a meshless method with enriched weight functions for three dimensional crack propagation the propagation of cracks in three dimensions is analysed by a meshless method the cracks are modelled by a set of triangles that are added when the propagation occurs since the method is meshless no remeshing of the domain is necessary during the propagation to avoid using a large number of degrees of freedom the stress singularity along the front of the cracks is taken into account by an enrichment of the shape functions of the meshless method by means of appropriate weight functions this enrichment technique is an extension of the technique that proved to be successful in two dimensions in a previous paper several algorithms for efficiently implementing the meshless method in three dimensions are detailed the accuracy of the enrichment is first assessed on simple examples and some results of non planar propagation of multiple cracks are then presented copyright 2005 john wiley sons ltd

existence and uniqueness results for third order nonlinear differential systems abstract sufficient conditions are given for the existence of solutions to third order boundary value problems for nonlinear differential systems some appropriate conditions are given to guarantee that the nagumo condition is satisfied by constructing appropriate a lower solution upper solution pair a concept that is defined in this paper the uniqueness result of the problem is also established the emphasis here is that the differential systems has nonlinear dependence on all over order derivatives and the boundary conditions are nonlinear

a computational method for solving singularly perturbed turning point problems exhibiting twin boundary layers singularly perturbed turning point problems tpps for second order ordinary differential equations des exhibiting twin boundary layers are considered in order to obtain numerical solution of these problems a computational method is suggested in which exponentially fitted difference schemes are combined with classical numerical methods in this method the given internal domain of definition of differential equation is divided into four subintervals and we solve the differential equation in each interval and combine the solution the implementation of the method on parallel architectures is mentioned error estimates are derived and numerical examples are given

a computational method for solving boundary value problems for third order singularly perturbed ordinary differential equations singularly perturbed two point boundary value problems spbvps for third order ordinary differential equations odes with a small parameter multiplying the highest derivative are considered a numerical method is suggested in this paper to solve such problems in this method the given bvp is transformed into a system of two odes subject to suitable initial and boundary conditions then the domain of definition of the differential equation a closed interval is divided into two sub intervals which we call the inner region boundary layer and the outer region then the de is solved in these intervals separately the solutions obtained in these intervals are combined to give the solution in the whole interval to obtain boundary conditions at the transition points boundary values inside this interval we use mostly the zeroth order asymptotic expansion of the solution of the bvp or a suitable asymptotic expansion solution first the linear equations are considered and then the semi linear equations to solve semi linear equations newton s method of quasi linearisation is applied examples are provided to illustrate the method the method is easy to implement and suitable for parallel computing

an asymptotic numerical fitted mesh method for singularly perturbed third order ordinary differential equations of reaction diffusion type singularly perturbed boundary value problems spbvps for third order ordinary differential equations odes with a small parameter multiplying the highest derivative of the form ey x b x y x c x y x f x y 0 p y 0 q y 1 r where b x c x and f x are sufficiently smooth functions satisfying certain conditions are considered firstly this third order singularly perturbed boundary value problem spbvp is transformed into equivalent problem of weakly coupled system of one first order and one second order ode with a small parameter multiplying the highest derivative of the second order ode subject to initial and boundary conditions respectively a computational method is suggested in this paper to solve this system in this method we first find the zero order asymptotic expansion approximation of the solution of the weakly coupled system then this system is decoupled by approximating the first component of the solution by its zero order asymptotic expansion approximation in the second equation finally the second equation is solved by the fitted mesh method j j h miller e o riordan g i shishkin in error estimates in the maximum norm for linear problems in one and two dimensions world scientific singapore 1996 numerical experiments are conducted

a computational method for solving singular perturbation problems abstract a class of singularly perturbed two point boundary value problems for second order ordinary differential equations is considered in order to solve them a computational method is suggested in which exponentially fitted difference schemes are combined with classical numerical methods the proposed method is distinguished by the following facts first we divide the given interval the domain of definition of differential equation into two subintervals called inner and outer regions then we solve the differential equation over both the regions as two point boundary value problems the terminal boundary condition of the inner region is obtained using the zero order asymptotic expansion of the solution some numerical examples are given to illustrate the method

an asymptotic numerical method for singularly perturbed third order ordinary differential equations of convection diffusion type abstract singularly perturbed two point boundary value problems spbvps for third order ordinary differential equations odes with a small parameter multiplying the highest derivative are considered a numerical method is suggested in this paper to solve such problems in this method the given bvp is transformed into a weakly coupled system of two odes subject to suitable initial and boundary conditions then the computational method presented in this paper is applied to this system in this method we reduce the weakly coupled system into a decoupled system then to solve this decoupled system numerically we apply a boundary value technique bvt in which the domain of definition of the differential equation is divided into two nonoverlapping subintervals called inner and outer regions then we solve the decoupled system over these regions as two point boundary value problems an exponentially fitted finite difference scheme is used in the inner region and a classical finite difference scheme in the outer region the boundary conditions at the transition point are obtained using the zero order asymptotic expansion approximation of the solution of the problem this computational method is distinguished by the facts that the decoupling reduces the computational time very much and it is well suited for parallel computing this method can be extended to a system of two ordinary differential equations of which one is of first order and the other is of second order numerical examples are given to illustrate the method

numerical treatment of singularly perturbed two point boundary value problems we propose a method for numerically solving linear singularly perturbed two point boundary value problems in ordinary differential equations with a boundary layer on the left end of the underlying interval this is a practical method and can be easily implemented on a computer the original problem is divided into inner and outer region differential equation systems the reduced problem is solved to obtain the terminal boundary condition then a new inner region problem is created and solved as a two point boundary value problem tpbvp in turn the outer region problem is also solved as a tpbvp both these tpbvps are efficiently treated by employing a slightly modified classical finite difference scheme coupled with discrete invariant imbedding algorithm to obtain the numerical solutions the stability of some recurrence relations involved in the algorithm is investigated the proposed method is iterative on the terminal point some numerical examples are included and the computational results are compared with exact solutions it is observed that the accuracy predicted can always be achieved with very little computational effort

a numerical method for singular perturbation problems arising in chemical reactor theory abstract a class of singularly perturbed two point boundary value problems for second order ordinary differential equations with mixed boundary conditions arising in chemical reactor theory is considered in order to solve them a numerical method is suggested in which an exponentially fitted difference scheme is combined with classical numerical methods the proposed method is distinguished by the following facts first we divide the given interval the domain of definition of the differential equation into two subintervals called outer and inner regions then we solve the differential equation over both the regions as two point boundary value problems the terminal boundary condition of the inner region is obtained using the zero order asymptotic expansion of the solution some numerical examples are given to illustrate the method

the asymptotic solution of a class of third order boundary value problems arising in the theory of thin film flows this paper studies boundary and interior layer phenomena exhibited by solutions of certain singularly perturbed third order boundary value problems which govern the motion of thin liquid films subject to viscous capillary and gravitational forces precise conditions specifying where and when the third order derivative terms in the differential equations can be neglected are derived and improved estimates for the actual solutions in terms of solutions of the lower order models are constructed the paper also contains a technique for replacing a third order problem with an asymptotically equivalent second order one that may have wider applicability

asymptotic numerical method for boundary value problems for singularly perturbed fourth order ordinary differential equations with a weak interior layer abstract singularly perturbed two point boundary value problems spbvps of convection diffusion type for fourth order ordinary differential equations odes with a small positive parameter multiplying the highest derivative with a discontinuous source term is considered the given fourth order bvp is transformed into a system of weakly coupled systems of two second order odes one without the parameter and the other with the parameter e multiplying the highest derivative and suitable boundary conditions in this paper a computational method for solving this system is presented in this method we first find the zero order asymptotic approximation expansion of the solution of the weakly coupled system then the system is decoupled by replacing the first component of the solution by its zero order asymptotic approximation expansion of the solution in the second equation then the second equation is solved by the numerical method which is constructed for this problem which involves shishkin mesh

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

singularly perturbed third order boundary value problem for nonlinear systems abstract this paper deals with the existence of solutions to a singularly perturbed boundary value problem for third order nonlinear differential systems the authors construct an appropriate generalized lower solution upper solution pair a concept that is defined in this paper and employ the method of descent nagumo conditions algebraic boundary layer functions and volterra type integral operator to guarantee the existence of solutions of the problem the uniformly valid asymptotic estimate of the solutions is given as well the differential systems has nonlinear dependence on all over order derivatives of the unknown

die gleichmaβige stabilitat singular gestorter diskreter und kontinuierlicher randwertprobleme in this paper we consider singular perturbation problems for ordinary differential operators of ordern and their discrete counterparts on arbitrary nonuniform grids we prove that the singularly perturbed initial value problem is stable uniformly in the perturbation parameter in both the continuous and the discrete case we use this result to characterize the stability of the corresponding continuous and discrete boundary value problems if the continuous problem is stable and if the consistency error is smaller than a certain constant the discrete problem is also stable

further examples of the boundary value technique in singular perturbation problems illustration de la technique des valeurs frontieres dans les problemes a perturbations singulieres par des exemples varies

singular perturbations of boundary value problems for a class of third order nonlinear ordinary differential equations abstract this article shows the existence uniqueness and asymptotic estimates of solutions of the singularly perturbed boundary value problems for a class of the third order nonlinear ordinary differential equations with a small parameter e 0 ey x y y e satisfying the condition y 0 0 y 1 0 y 1 0 or y 0 0 y 1 0 y 1 0

computational methods for solving two parameter singularly perturbed boundary value problems for second order ordinary differential equations singularly perturbed two point boundary value problems spbvps for second order ordinary differential equations odes with two small parameters multiplying the derivatives are considered computational methods are suggested in this paper to solve this type of problems one of these methods is fitted operator method fom and the other is bondary value technique bvt in fom we suggest an exponentially fitted finite difference effd scheme throughout the domain of definition of the differential equation and the error estimates are derived in bvt the domain of definition of the differential equation is divided into three subintervals called inner and outer regions then we solve the two parameter spbvp tpspbvp over these regions as two point boundary value problems an exponentially fitted finite difference scheme is used in the inner regions and a classical finite difference scheme in the outer region the data for boundary conditions at the transition points are obtained using zero order asymptotic expansion approximation of the solution of the problem the bvt is well suited for parallel computing numerical examples are given to illustrate the method

asymptotic numerical methods for singularly perturbed fourth order ordinary differential equations of convection diffusion type singularly perturbed boundary value problems bvps for fourth order ordinary differential equations odes with a small positive parameter multiplying the highest derivative of the email protected i v x a x email protected x b x y x c x y x f x email protected d 0 1 y 0 p y 1 q y 0 r y 1 s are considered the given fourth order bvp is transformed into a system of weakly coupled system of two second order odes one without the parameter and the other with the parameter e multiplying the highest derivative and suitable boundary conditions in this paper computational methods for solving this system are presented in these methods we first find the zero order asymptotic approximation expansion of the solution of the weakly coupled system then the system is decoupled by replacing the first component of the solution by its zero order asymptotic approximation expansion of the solution in the second equation then the second equation is solved by the fitted operator method fom fitted mesh method fmm and boundary value technique bvt error estimates are derived and examples are provided to illustrate the methods

non planar 3d crack growth by the extended finite element and level sets part ii level set update we present a level set method for treating the growth of non planar three dimensional cracks the crack is defined by two almost orthogonal level sets signed distance functions one of them describes the crack as a two dimensional surface in a three dimensional space and the second is used to describe the one dimensional crack front which is the intersection of the two level sets a hamilton jacobi equation is used to update the level sets a velocity extension is developed that preserves the old crack surface and can accurately generate the growing surface the technique is coupled with the extended finite element method which approximates the displacement field with a discontinuous partition of unity this displacement field is constructed directly in terms of the level sets so the discretization by finite elements requires no explicit representation of the crack surface numerical experiments show the robustness of the method both in accuracy and in treating cracks with significant changes in topology copyright 2002 john wiley sons ltd

the role of linear objective functions in barrier methods we consider the asymptotic behavior of the newton log barrier method for inequality constrained optimization we show that when the objective function is linear an effective step can be taken along the newton direction after each reduction in the barrier parameter leading to efficient performance during the final stages of the algorithm this behavior contrasts with the case of a nonlinear objective where newton s method often performs more and more poorly as the barrier parameter is reduced to zero we analyze the behavior and demonstrate our result on a simple example

a rational spectral collocation method for third order singularly perturbed problems a new numerical method is developed for solving a class of third order singularly perturbed boundary value problems first of all the given problem is transformed into a system of two ordinary differential equations odes subject to suitable initial and boundary conditions then the rational spectral collocation method in barycentric form with sinh transformation is applied to solve the system of odes according to the asymptotic analysis the location and width of boundary layer of the given problem which are chosen as parameters in the sinh transformation can be determined ample numerical experiments are presented to illustrate the computational efficiency and accuracy of the our method

numerical solution of singular perturbation problems via deviating arguments we present a new approach to numerically solving linear singularly perturbed two point boundary value problems in ordinary differential equations with a boundary layer on the left end of the interval the original problem is divided into outer and inner region problems a terminal boundary condition in the implicit form is derived then the outer region problem is solved as a two point boundary value problem tpbvp and an explicit terminal boundary condition is obtained in turn a new inner region problem is obtained and solved as a tpbvp using the explicit terminal boundary condition the proposed method is iterative on the terminal point of the inner region some numerical examples have been solved to demonstrate the applicability of the method

singular perturbations of third order nonlinear differential equations with full nonlinear boundary conditions in this paper we discuss singular perturbations of third order nonlinear ordinary differential equations with full nonlinear boundary conditions the emphasis here is that the nonlinear term depends on the first second order derivatives and the boundary conditions are full nonlinear that is where the main novelty of this work lies by applying the upper and lower solutions method as well as analysis technique the existence uniqueness results for the singularly perturbed boundary value problem are established and asymptotic estimates of solutions is also obtained