interaction of an edge dislocation with a coated elliptic inclusion this paper presents an analytical solution for plane elasticity problems of an elliptically cylindrical layered media subject to an arbitrary edge dislocation based on the technique of conformal mapping and the method of analytical continuation in conjunction with the alternating technique the general expressions of the displacements and stresses where an edge dislocation is located in matrix coating layer and inclusion are obtained the numerical results of image forces exerted on a generalized edge dislocation are carried out by using the generalized peach koehler equation as a numerical illustration both the image forces and equilibrium positions are presented for different material combinations and relative thickness of a coating layer the result shows that the thickness and the shear modulus of the coating layer have a strong influence on the stability of dislocation

dual hybrid trefftz element formulation based on independent boundary traction frame reference epfl article 102727view record in web of science record created on 2007 04 23 modified on 2016 08 08

applications of the generalized singular value decomposition method on the eigenproblem using the incomplete boundary element formulation in this paper four incomplete boundary element formulations including the real part singular boundary element the real part hypersingular boundary element the imaginary part boundary element and the plane wave element methods are used to solve the free vibration problem among these incomplete boundary element formulations the real part singular and the hypersingular boundary elements are of the singular type and the other two are of the regular type when the incomplete formulation is used the spurious eigensolution may be encountered an auxiliary system whose boundary conditions are linearly independent of those of the original system is required in the proposed method in order to eliminate the spurious eigensolution a mathematical proof is given to show that the spurious eigensolution will appear in both the original and auxiliary systems as a result one can eliminate spurious eigensolution by means of the generalized singular value decomposition method in addition to the spurious eigensolution problem the regular boundary element formulation further suffers ill conditioned behaviors in the problem solver it is explained analytically using a circular domain why ill conditioned behaviors exist two main ill conditioned problems are the numerical instability of solution and the nonexistence of solution to solve the numerical instability of solution problem existing in these proposed regular formulations the tikhonov s regularization technique is used to improve the condition number of the leading coefficient matrix further the generalized singular value decomposition method is used to eliminate spurious eigensolutions numerical examples are given to verify the performance of these proposed methods and the results match the analytical values well

dual boundary element method simple error estimator and adaptivity this paper is concerned with the effective numerical implementation of the adaptive dual boundary element method dbem for two dimensional potential problems two boundary integral equations which are the potential and the flux equations are applied for collocation along regular and degenerate boundaries leading always to a single region analysis taking advantage on the use of non conforming parametric boundary elements the method introduces a simple error estimator based on the discontinuity of the solution across the boundaries between adjacent elements and implements the p h and mixed versions of the adaptive mesh refinement examples of several geometries which include degenerate boundaries are analyzed with this new formulation to solve regular and singular problems the accuracy and efficiency of the implementation described herein make this a reliable formulation of the adaptive dbem

applications of the direct trefftz boundary element method to the free vibration problem of a membrane in this paper the direct trefftz method is applied to solve the free vibration problem of a membrane in the direct trefftz method there exists no spurious eigenvalue however an ill posed nature of numerical instability encountered in the direct trefftz method requires some treatments the tikhonov s regularization method and generalized singular value decomposition method are used to deal with such an ill posed problem numerical results show the validity of the current approach

variationally consistent extended fe model for 3d planar and curved imperfect interfaces abstract we propose an extended finite element method convergent to the asymptotic solution of a thin interface problem for both planar and curved imperfect interfaces in three dimensions the main advantage over standard cohesive zone models is the bulk mesh size independence with respect to standard extended finite element method in the proposed procedure blending and quadrature sub domains are not required the focus is on the evaluation of the accuracy of the proposed approach in solving three dimensional benchmark tests the numerical results are compared with those available from analytical solutions and spring like interface models

application of the trefftz method in plane elasticity problems direct and indirect approximations using sets of non singular complete trefftz functions i e the complete systems of solutions have been successfully applied to harmonic problems 1 in this paper the procedure is applied to a more complex situation plane elasticity problems the examples show that the present method can avoid the difficulties relating to singular integration and has good accuracy compared with traditional boundary elements

integration of singular enrichment functions in the generalized extended finite element method for three dimensional problems a mapping method is developed to integrate weak singularities which result from enrichment functions in the generalized extended finite element method the integration scheme is applicable to 2d and 3d problems including arbitrarily shaped triangles and tetrahedra implementation of the proposed scheme in existing codes is straightforward numerical examples for 2d and 3d problems demonstrate the accuracy and convergence properties of the technique copyright 2008 john wiley sons ltd

analytical solution for a reinforcement layer bonded to an elliptic hole under a remote uniform load a general solution to a reinforced elliptic hole embedded in an infinite matrix subjected to a remote uniform load is provided in this paper investigations on the present elasticity problem are rather tedious due to the presence of material inhomogeneities and complex geometric configurations based on the technique of conformal mapping and the method of analytical continuation in conjunction with the alternating technique the general expressions of the displacement and stresses in a reinforcement layer and the matrix are derived explicitly in a series form some numerical results are provided to investigate the effects of the material combinations and geometric configurations on the interfacial stresses the results show that there exists an optimum design of a reinforcement layer such that both the magnitude of stress concentration and the interfacial stresses could be fairly reduced

highly accurate surface and volume integration on implicit domains by means of moment fitting summary we introduce a new method for the numerical integration over curved surfaces and volumes defined by a level set function the method is based on the solution of a small linear system based on a simplified variant of the moment fitting equations numerical experiments suggest that the accuracy of the resulting quadrature rules exceeds the accuracy of traditional methods by orders of magnitude using moments up to an order of p the measured experimental orders of convergence exceed hp consequently their construction is very efficient because only coarse computational grids are required the conceptual simplicity allows for the application on very general grid types which is demonstrated by numerical experiments on quadrilateral triangular and hexahedral grids copyright 2013 john wiley sons ltd

application of symmetric indirect trefftz method to free vibration problems in 2d a symmetric indirect trefftz method is developed to solve the free vibration problem of a 2d membrane it is proved that in this approach the spurious eigensolution exists and an auxiliary matrix is constructed to help extraction of the spurious solution using the generalized singular value decomposition in addition to the spurious eigensolution this regular formulation suffers from its ill posed nature i e the numerical instability in order to deal with the numerical instability the tikhonov s regularization method in conjunction with the generalized singular value decomposition is suggested the proposed approach has some merits when compared with other regular boundary element formulations reported so far namely the capacity of representing eigenmodes and the ability to deal with a multiply connected domain of genus 1 several numerical examples are demonstrated to show the validity of the current approach copyright 2003 john wiley sons ltd

the extended generalized finite element method an overview of the method and its applications an overview of the extended generalized finite element method gefm xfem with emphasis on methodological issues is presented this method enables the accurate approximation of solutions that involve jumps kinks singularities and other locally non smooth features within elements this is achieved by enriching the polynomial approximation space of the classical finite element method the gefm xfem has shown its potential in a variety of applications that involve non smooth solutions near interfaces among them are the simulation of cracks shear bands dislocations solidification and multi field problems copyright 2010 john wiley sons ltd

trefftz method for kirchhoff plate bending problems kirchhoff plate bending problems were studied by both the trefftz direct and indirect approximations in which non singular complete trefftz functions are used as the weighting and or trial functions the trefftz direct method involved only the quantities of engineering interest numerical results are given to show the efficiency and the excellent accuracy of the present method

higher order extended fem for weak discontinuities level set representation quadrature and application to magneto mechanical problems in this article we present the application of bilinear and biquadratic extended fem xfem formulations to model weak discontinuities in magnetic and coupled magneto mechanical boundary value problems for properly resolving the location of curved interfaces and the discontinuous physical behaviour the major part of the contribution is devoted to review and develop methods for level set representation of curved interfaces and numerical integration of the weak form in higher order xfem formulations in order to reduce the complexity of the representation of curved interfaces an element local approach that allows for an automated computation of the level set values and also improves the compatibility between the level set representation and the integration subdomains is proposed integration rules for polygons and strain smoothing are applied in conjunction with biquadratic elements and compared with curved integration subdomains eventually a coupled magneto mechanical demonstration problem is modelled and solved by xfem for demonstration purposes a magneto mechanical coupling due to magnetic stresses is considered errors and convergence rates are analysed for the different level set representations and numerical integration procedures as well as their dependence on the ratio of material parameters at an interface copyright c 2012 john wiley sons ltd

fast integration and weight function blending in the extended finite element method two issues in the extended finite element method xfem are addressed efficient numerical integration of the weak form when the enrichment function is self equilibrating and blending of the enrichment the integration is based on transforming the domain integrals in the weak form into equivalent contour integrals it is shown that the contour form is computationally more efficient than the domain form especially when the enrichment function is singular and or discontinuous a method for alleviating the errors in the blending elements is also studied in this method the enrichment function is pre multiplied by a smooth weight function with compact support to allow for a completely smooth transition between enriched and unenriched subdomains a method for blending step function enrichment with singular enrichments is described it is also shown that if the enrichment is not shifted properly the weighted enrichment is equivalent to the standard enrichment an edge dislocation and a crack problem are used to benchmark the technique the influence of the variables that parameterize the weight function is analyzed the resulting method shows both improved accuracy and optimum convergence rates and is easily implemented into existing xfem codes

the hybrid trefftz finite element model and its application to plate bending this paper presents a new hybrid element approach and applies it to plate bending in contrast to more conventional models the formulation is based on displacement fields which fulfil a priori the non homogeneous lagrange equation trefftz method the interelement continuity is enforced by using a stationary principle together with an independent interelement displacement the final unknowns are the nodal displacements and the elements may be implemented without any difficulty in finite element libraries of standard finite element programs the formulation only calls for integration along the element boundaries which enables arbitrary polygonal or even curve sided elements to be generated where relevant known local solutions in the vicinity of a singularity or stress concentration may be used as an optional expansion basis to obtain for example particular singular corner elements elements presenting circular holes etc thus a high degree of accuracy may be achieved without a troublesome mesh refinement another important advantage of the formulation is the possibility of generating by a single element subroutine a large number of various elements triangles quadrilaterals etc presenting an increasing degree of accuracy the paper summarizes the results of numerical studies and shows the excellent accuracy and efficiency of the new elements the conclusions present some ideas concerning the adaptive version of the new elements extension to nonlinear problems and some other developments

equivalent polynomials for quadrature in heaviside function enriched elements summary one of the advantages of partition of unity fems like the extended fem is the ability of modeling discontinuities independent of the mesh structure the enrichment of the element functional space with discontinuous or non differentiable functions requires when the element stiffness is computed partitioning into subdomains for quadrature however the arbitrary intersection between the base mesh and the discontinuity plane generates quadrature subdomains of complex shape this is particularly true in three dimensional problems where quite sophisticate methodologies have been presented in the literature for the element stiffness evaluation the present work addresses the problem of heaviside function enrichments and is based on the replacement of the discontinuous enrichment function with the limit of an equivalent polynomial defined on the entire element domain this allows for the use of standard gaussian quadrature in the elements crossed by the discontinuity the work redefines conceptually the first version of the equivalent polynomial methodology introduced in 2006 allowing a much broader applicability as a consequence equivalent polynomials can be computed for all continuum element families in one two and three dimensions copyright 2014 john wiley sons ltd

an asymmetric indirect trefftz method for solving free vibration problems in this paper a new asymmetric indirect trefftz method aitm has been developed to solve free vibration problems the proposed method is categorized into a regular type of boundary element methods bems such that no singular or hypersingular integration is necessary however like other regular bems the proposed approach encounters the numerical instability as the number of elements increases to deal with such an ill posed behavior tikhonov s regularization method in conjunction with the generalized singular value decomposition gsvd is adopted it is proved that the degeneracy of the proposed indirect trefftz method has the same mathematical structure as the direct trefftz method thus no special effort should be paid in programming besides such an equivalency indicates that the current method does not have spurious eigensolutions furthermore the proposed approach can easily treat a multiply connected domain of genus 1 as shown in fig 1 due to its indirect nature the present approach can also represent the mode shapes within its own mathematical formulations several numerical examples are given to show the validity of the proposed approach

an application of trefftz method to the sensitivity analysis of two dimensional potential problem this paper presents a new formulation for the sensitivity analysis with the help of the trefftz boundary type solution procedure direct differentiation of the expressions of the field variables leads to the expressions of the sensitivities since the field variables are approximated by linear combination of regular t complete functions the analytical expressions of the sensitivities are also regular sensitivity analysis schemes with respect to variations of dimensions of the object and boundary conditions are explained in detail two dimensional potential problems in the bounded region are considered as a numerical example in order to confirm its validity

generalized gaussian quadrature rules for discontinuities and crack singularities in the extended finite element method new gaussian integration schemes are presented for the efficient and accurate evaluation of weak form integrals in the extended finite element method for discontinuous functions we construct gauss like quadrature rules over arbitrarily shaped elements in two dimensions without the need for partitioning the finite element a point elimination algorithm is used in the construction of the quadratures which ensures that the final quadratures have minimal number of gauss points for weakly singular integrands we apply a polar transformation that eliminates the singularity so that the integration can be performed efficiently and accurately numerical examples in elastic fracture using the extended finite element method are presented to illustrate the performance of the new integration techniques

improving the accuracy of xfem crack tip fields using higher order quadrature and statically admissible stress recovery this study is concerned with improving the accuracy of crack tip fields obtained using the extended generalized finite element method xfem first the numerical integration of the element stiffness matrices which guarantees convergence with quadrature of not only the regular nodal displacements but also additional degrees of freedom corresponding to the enrichment functions is studied as the accuracy of the stresses obtained by direct differentiation of the converged with quadrature regular nodal displacements and of the coefficients corresponding to enrichment functions is still not satisfactory a statically admissible stress recovery sar scheme is introduced sar uses basis functions which meet the equilibrium equations within the domain and the local traction conditions on the boundary and moving least squares mls to fit the stresses at sampling points e g quadrature points obtained by the xfem important parameters controlling the accuracy of crack tip fields using the xfem and sar namely the order of quadrature the number of retained terms in the crack tip asymptotic field the number of enriched layers and use of arbitrary branch functions a proper choice of the sampling points in the enriched element and the size of the domain of influence doi of mls are investigated copyright 2005 john wiley sons ltd

the design and analysis of the generalized finite element method abstract in this paper we introduce the generalized finite element method gfem as a combination of the classical finite element method fem and the partition of unity method pum the standard finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data e g the singular functions obtained from the local asymptotic expansion of the exact solution in the neighborhood of a corner point etc the special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and pasted together with the existing finite element basis to construct an augmented conforming finite element space in this way the local approximability afforded by the special functions is included in the approximation while maintaining the existing infrastructure of finite element codes the major features of the gfem are 1 the essential boundary conditions can be imposed exactly as in the standard fem unlike other partition of unity based methods where this is a major issue 2 the accuracy of the numerical integration of the entries of the stiffness matrix and load vector is controlled adaptively so that the errors in integration of the special functions do not affect the accuracy of the constructed approximation this issue also has not been sufficiently addressed in other implementations of partition of unity or meshless methods and 3 linear dependencies in the system of equations are resolved by employing an easy modification of the direct linear solver the power of the gfem for solving problems in domains with complex geometry with less error and less computer resources than the standard fem is illustrated by numerical examples

on the equivalence of the trefftz method and method of fundamental solutions for laplace and biharmonic equations in this paper it is proved that the two approaches known in the literature as the method of fundamental solutions mfs and the trefftz method are mathematically equivalent in spite of their essentially minor and apparent differences in formulation in deriving the equivalence of the trefftz method and the mfs for the laplace and biharmonic problems it is interesting to find that the complete set in the trefftz method for the laplace and biharmonic problems are embedded in the degenerate kernels of the mfs the degenerate scale appears using the mfs when the geometrical matrix is singular the occurring mechanism of the degenerate scale in the mfs is also studied by using circulant the comparison of accuracy and efficiency of the two methods was addressed

quadrature schemes for arbitrary convex concave volumes and integration of weak form in enriched partition of unity methods abstract quadrature schemes are constructed based on moment fitting equations to integrate polynomials over arbitrary convex concave volumes that arise among others in enriched partition of unity finite element methods epum the building block of the scheme involves the divergence theorem of multivariable calculus which is used to integrate the base functions an efficient and robust point distribution method is proposed and the quadrature weights at the corresponding points are obtained by solving a least squares problem the method is applied initially to integrate given polynomial functions over complex volumes and further to simulate simple three dimensional fluid dynamic problems which involve very complex volumes when solved with epum accuracy of the present quadrature construction scheme is demonstrated by comparing the results with the available exact numerical solutions and efficiency of the method is proved by comparing the computational time with that of the widely used tessellation method

circularly cylindrical layered media in plane elasticity this paper presents an alternative efficient procedure to analyze plane elasticity problems of a circularly cylindrical layered media subject to an arbitrary point force based on the method of analytical continuation in conjunction with the alternating technique the elastic fields of the three phase media are derived a rapidly convergent series solution which is expressed in terms of an explicit general term of the complex potential of the corresponding homogeneous problem is obtained in an elegant form as a numerical illustration the interfacial stresses are presented for different material combinations and for different positions of the point force

trefftz indirect methods for plane piezoelectricity in this paper a unified theory for the general solutions of three dimensional elasticity is employed to transform the governing equations of plane piezoelectricity into three generalized laplace equations the latter equations are uncoupled and can be pursued separately by introducing three generalized complex variables the complete solution sets for plane piezoelectricity are constructed by adopting the constituents in the solution sets as the trial functions the trefftz collocation and the trefftz galerkin methods are formulated both methods fall into the category of trefftz indirect methods as the coefficients of their trial functions are non physical numerical examples are presented to illustrate the efficacy of the formulations copyright 2005 john wiley sons ltd

a regularized xfem model for the transition from continuous to discontinuous displacements this work focuses on the modelling through the extended finite element method of structural problems characterized by discontinuous displacement as a model problem an elastic isotropic domain characterized by a displacement discontinuity across a surface is studied a regularization of the displacement field is introduced depending on a scalar parameter the regularized solution is defined in a layer the emerging strain and stress fields are independently modelled using specific constitutive assumptions in particular it is shown that the mechanical work spent within the regularization layer can be interpreted as an interface work provided that a spring like constitutive law is adopted the accuracy of the integration procedures adopted for the stiffness matrix is assessed as highly non linear terms appear standard gauss quadrature is compared with adaptive quadrature and with a new technique based on an equivalent polynomial approach one and two dimensional results are reported for varying discretization size regularization parameter and constitutive parameters copyright 2007 john wiley sons ltd

generalized finite element analysis with t complete boundary solution functions the use of a complete and nonsingular set of trefftz functions in the solution of quasi harmonic equations is demonstrated and shown to be often superior to the more conventional singularity distribution in boundary type approximation procedures for coupling separate domains of such solution and indeed of deriving equivalent finite elements are demonstrated

singularities interfaces and cracks in dissimilar anisotropic media for a non pathological bimaterial in which an interface crack displays no oscillatory behaviour it is observed that apart possibly from the stress intensity factors the structure of the near tip field in each of the two blocks is independent of the elastic moduli of the other block collinear interface cracks are analysed under this non oscillatory condition and a simple rule is formulated that allows one to construct the complete solutions from mode iii solutions in an isotropic homogeneous medium the general interfacial crack tip field is found to consist of a two dimensional oscillatory singularity and a one dimensional square root singularity a complex and a real stress intensity factors are proposed to scale the two singularities respectively owing to anisotropy a peculiar fact is that the complex stress intensity factor scaling the oscillatory fields however defined does not recover the classical stress intensity factors as the bimaterial degenerates to be non pathological collinear crack problems are also formulated in this context and a strikingly simple mathematical structure is identified interactive solutions for singularity interface and singularity interface crack are obtained the general results are specialized to decoupled antiplane and in plane deformations for this important case it is found that if a material pair is non pathological for one set of relative orientations of the interface and the two solids it is non pathological for any set of orientations for bonded orthotropic materials an intuitive choice of the principal measures of elastic anisotropy and dissimilarity is rationalized a complex variable representation is presented for a class of degenerate orthotropic materials throughout the paper the equivalence of the lekhnitskii and stroh formalisms is emphasized the article concludes with a formal statement of interfacial fracture mechanics for anisotropic solids

high order extended finite element method for cracked domains the aim of the paper is to study the capabilities of the extended finite element method xfem to achieve accurate computations in non smooth situations such as crack problems although the xfem method ensures a weaker error than classical nite element methods the rate of convergence is not improved when the mesh parameter h is going to zero because of the presence of a singularity the di culty can be overcome by modifying the enrichment of the nite element basis with the asymptotic crack tip displacement solutions as well as with the heaviside function numerical simulations show that the modi ed xfem method achieves an optimal rate of convergence i e like in a standard nite element method for a smooth problem

dislocations by partition of unity a new finite element method for accurately modelling the displacement and stress fields produced by a dislocation is proposed the methodology is based on a local enrichment of the finite element space by closed form solutions for dislocations in infinite media via local partitions of unity this allows the treatment of both arbitrary boundary conditions and interfaces between materials the method can readily be extended to arrays of dislocations 3d problems large strains and non linear constitutive models results are given for an edge dislocation in a hollow cylinder and in an infinite medium for the cases of a glide plane intersecting a rigid obstacle and an interface between two materials copyright 2005 john wiley sons ltd

integrating strong and weak discontinuities without integration subcells and example applications in an xfem gfem framework partition of unity methods such as the extended finite element method allows discontinuities to be simulated independently of the mesh int j numer meth engng 1999 45 601 620 this eliminates the need for the mesh to be aligned with the discontinuity or cumbersome re meshing as the discontinuity evolves however to compute the stiffness matrix of the elements intersected by the discontinuity a subdivision of the elements into quadrature subcells aligned with the discontinuity is commonly adopted in this paper we use a simple integration technique proposed for polygonal domains int j nuttier meth engng 2009 80 1 103 134 doi 10 1002 nme 2589 to suppress the need for element subdivision numerical results presented for a few benchmark problems in the context of linear elastic fracture mechanics and a multi material problem show that the proposed method yields accurate results owing to its simplicity the proposed integration technique can be easily integrated in any existing code copyright c 2010 john wiley sons ltd

quadrature rules for triangular and tetrahedral elements with generalized functions quadrature rules are developed for exactly integrating products of polynomials and generalized functions over triangular and tetrahedral domains these quadrature rules greatly simplify the implementation of finite element methods that involve integrals over volumes and interfaces that are not coincident with the element boundaries specifically the integrands considered here consist of a quadratic polynomial multiplied by a heaviside or dirac delta function operating on a linear polynomial this form allows for exact integration of expressions obtained from linear finite elements over domains and interfaces defined by a linear level set function exact quadrature rules are derived that involve fixed quadrature point locations with weights that depend continuously on the nodal level set values compared with methods involving explicit integration over subdomains the quadrature rules developed here accommodate degenerate interface geometries without any need for special consideration and provide analytical jacobian information describing the dependence of the integrals on the nodal level set values the accuracy of the method is demonstrated for a simple conduction problem with the neumann and robin type boundary conditions copyright 2007 john wiley sons ltd

solution of helmholtz equation by trefftz method the application of the trefftz method for calculating wave forces on offshore structures is presented indirect and direct formulations using complete and non singular systems of trefftz functions for the helmholtz equation are posed in this paper an effective technique using different interpolation functions for the velocity potential and wave force are suggested to improve the computational accuracy of the wave force the numerical examples show that the present method is highly efficient and accurate