a three dimensional model describing stress temperature induced solid phase transformations solution algorithm and boundary value problems summary an always increasing knowledge on material properties as well as a progressively more sophisticated production technology make shape memory alloys sma extremely interesting for the industrial world at the same time sma devices are typically characterized by complex multi axial stress states as well as non homogeneous and non isothermal conditions both in space and time this aspect suggests the finite element method as a useful tool to help and improve application design and realization with this aim we focus on a three dimensional macroscopic thermo mechanical model able to reproduce the most significant sma features int j numer methods eng 2002 55 1255 1264 proposing a simple modification of such a model however the suggested modification allows the development of a time discrete solution algorithm which is more effective and robust than the one previously discussed in the literature we verify the computational tool ability to simulate realistic mechanical boundary value problems with prescribed temperature dependence studying three sma applications a spring actuator a selfexpanding stent a coupling device for vacuum tightness the effectiveness of the model to solve thermo mechanical coupled problems will be discussed in a forthcoming work copyright 2004 john wiley sons ltd

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

the xfem for high gradient solutions in convection dominated problems convection dominated problems typically involve solutions with high gradients near the domain boundaries boundary layers or inside the domain shocks the approximation of such solutions by means of the standard finite element method requires stabilization in order to avoid spurious oscillations however accurate results may still require a mesh refinement near the high gradients herein we investigate the extended finite element method xfem with a new enrichment scheme that enables highly accurate results without stabilization or mesh refinement a set of regularized heaviside functions is used for the enrichment in the vicinity of the high gradients different linear and non linear problems in one and two dimensions are considered and show the ability of the proposed enrichment to capture arbitrary high gradients in the solutions copyright c 2009 john wiley sons ltd

efficient adaptive integration of functions with sharp gradients and cusps in n dimensional parallelepipeds summary in this paper we study the efficient numerical integration of functions with sharp gradients and cusps an adaptive integration algorithm is presented that systematically improves the accuracy of the integration of a set of functions the algorithm is based on a divide and conquer strategy and is independent of the location of the sharp gradient or cusp the error analysis reveals that for a c0 function derivative discontinuity at a point a rate of convergence of n 1 is obtained in rn two applications of the adaptive integration scheme are studied first we use the adaptive quadratures for the integration of the regularized heaviside function a strongly localized function that is used for modeling sharp gradients then the adaptive quadratures are employed in the enriched finite element solution of the all electron coulomb problem in crystalline diamond the source term and enrichment functions of this problem have sharp gradients and cusps at the nuclei we show that the optimal rate of convergence is obtained with only a marginal increase in the number of integration points with respect to the pure finite element solution with the same number of elements the adaptive integration scheme is simple robust and directly applicable to any generalized finite element method employing enrichments with sharp local variations or cusps in n dimensional parallelepiped elements copyright 2012 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

a corrected xfem approximation without problems in blending elements the extended finite element method xfem enables local enrichments of approximation spaces standard finite elements are used in the major part of the domain and enriched elements are employed where special solution properties such as discontinuities and singularities shall be captured in elements that blend the enriched areas with the rest of the domain problems arise in general these blending elements often require a special treatment in order to avoid a decrease in the overall convergence rate a modification of the xfem approximation is proposed in this work the enrichment functions are modified such that they are zero in the standard elements unchanged in the elements with all their nodes being enriched and varying continuously in the blending elements all nodes in the blending elements are enriched the modified enrichment function can be reproduced exactly everywhere in the domain and no problems arise in the blending elements the corrected xfem is applied to problems in linear elasticity and optimal convergence rates are achieved copyright 2007 john wiley sons ltd

delta function approximations in level set methods by distance function extension in a k tornberg b engquist numerical approximations of singular source terms in differential equations j comput phys 200 2004 462 488 it was shown for simple examples that the then m

waring like decompositions of polynomials 1 abstract let f be a homogeneous form of degree d in n variables a waring decomposition of f is a way to express f as a sum of d th powers of linear forms in this paper we consider the decompositions of a form as a sum of expressions each of which is a fixed monomial evaluated at linear forms

nonlinear thermomechanical oscillations of shape memory devices the nonlinear responses and bifurcations of shape memory oscillators based on a thermomechanical model are investigated employing a numerically refined approach because of the discontinuities of the system tangent stiffness classical gradient based shooting type techniques for calculating the periodic responses are not applicable these solutions are then sought as the fixed points of the poincarmap combined with a path following procedure the jacobian of the map is calculated via a central finite difference scheme and its eigenvalues the floquet multipliers are computed to ascertain the stability of the solutions and the codimension one bifurcations frequency response curves are constructed for shape memory oscillators characterized by different hysteresis loops and for various excitation levels the investigations are conducted both in isothermal and nonisothermal conditions and the main outcomes are comparatively discussed a rich class of solutions and bifurcations including jump phenomena pitchfork period doubling hopf bifurcations complete bubble structures culminating into chaos is found quasiperiodic motions arise in nearly adiabatic conditions 2003 elsevier ltd all rights reserved

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

on the waring problem for polynomial rings in this note we discuss an analog of the classical waring problem for namely we show that a general homogeneous polynomial of degree divisible by k 2 can be represented as a sum of at most kn k th powers of homogeneous polynomials in noticeably kn coincides with the number obtained by naive dimension count

on the secant varieties to the tangent developable of a veronese variety abstract let v n d p n for n n d n 1 be the order d veronese embedding of p n x n d t v n d p n the tangent developable of v n d and s s 1 x n d p n the s secant variety of x n d i e the closure in p n of the union of all s 1 linear spaces spanned by s points of x n d s s 1 x n d has expected dimension min n 2 n 1 s 1 catalisano geramita and gimigliano conjectured that s s 1 x n d has always the expected dimension except when d 2 n 2 s or d 3 and n 2 3 4 in this paper we prove their conjecture when n 2 and n 3

numerical algorithms for cyclic phase transformation hysteresis in a shape memory plate subject to axisymmetric plane stress we present numerical algorithms for calculating stress fields in an annulus composed of a shape memory material under conditions of quasi static edge loading at constant temperature the algorithms track the material microstructure in terms of the volume fraction of austenite a and martensite m the latter of which provides a transformation strain the dependence on load path imparts significant hysteresis in the stress induced transformation between a and m a previous study that was restricted to proportional loading in the direction of forward transformation j appl mech 2005 72 44 53 is here generalized to consider arbitrary loadings the shooting algorithm that was robust for the previously considered proportional loadings is found to be subject to numerical instability for the most general transformation possibilities considered here this motivates the development of an alternative iterated mapping algorithm that is found to generate a robust semi analytical finite difference procedure the algorithm efficiently determines the operative transformation type as is illustrated in cases where forward and reverse loading are occurring simultaneously at different plate locations at those locations where phase transformation is inactive the algorithm continues to account for martensite reorientation that alters the local transformation strain copyright 2006 john wiley sons ltd

a brief proof of a maximal rank theorem for generic double points in projective space we give a simple proof of the following theorem of j alexander and a hirschowitz given a general set of points in projective space the homogeneous ideal of polynomials that are singular at these points has the expected dimension in each degree of 4 and higher except in 3 cases

most secant varieties of tangential varieties to veronese varieties are nondefective we prove a conjecture stated by catalisano geramita and gimigliano in 2002 which claims that the secant varieties of tangential varieties to the dth veronese embedding of the projective n space pn have the expected dimension modulo a few wellknown exceptions it is arguably the first complete result on the dimensions of secant varieties of a classically studied variety since the work of alexander and hirschowitz in 1995 as bernardi catalisano gimigliano and ida demonstrated that the proof of this conjecture may be reduced to the case of cubics i e d 3 the main contribution of this work is the resolution of this base case the proposed proof proceeds by induction on the dimension n of the projective space via a specialization argument this reduces the proof to a large number of initial cases for the induction which were settled using a computer assisted proof the individual base cases were computationally challenging problems indeed the largest base case required us to deal with the tangential variety to the third veronese embedding of p79 in p88559

secant varieties to osculating varieties of veronese embeddings of p n a well known theorem by alexander hirschowitz states that all the higher secant varieties of v n d the d uple embedding of pp n have the expected dimension with few known exceptions we study here the same problem for t n d the tangential variety to v n d and prove a conjecture which is the analogous of alexander hirschowitz theorem for n leq 9 moreover we prove that it holds for any n d if it holds for d 3 then we generalize to the case of o k n d the k osculating variety to v n d proving for n 2 a conjecture that relates the defectivity of sigma s o k n d to the hilbert function of certain sets of fat points in pp n

3d corrected xfem approach and extension to finite deformation theory in this paper the modified or corrected extended finite element method originally presented in fries int j numer meth engng 2008 75 503 532 for the 2d case is extended to 3d including different remedies for the problem that the crack front enrichment functions are linearly dependent in the blending elements in the context of this extension we address a number of computational issues of the 3d xfem in particular possible quadrature rules for elements with discontinuities also the influence of finite deformation theory for crack simulations in comparison to linear elastic fracture mechanics is investigated a number of numerical examples demonstrate the behavior of the presented possibilities copyright 2010 john wiley sons ltd

analysis of a one dimensional model for the immersed boundary method numerical methods are studied for the one dimensional heat equation with a singular forcing term u t u xx c t delta x alpha t the delta function delta x is replaced by a discrete approximation d h x and the resulting equation is solved by a crank nicolson method on a uniform grid the accuracy of this method is analyzed for various choices of d h the case where c t is specified and also the case where c is determined implicitly by a constraint on the solution at the point a are studied these problems serve as a model for the immersed boundary method of peskin for incompressible flow problems in irregular regions some insight is gained into the accuracy that can be achieved and the importance of choosing appropriate discrete delta functions

accuracy of three dimensional analysis of regularized singularities summary in computational mechanics the quadrature of discontinuous and singular functions is often required to avoid specialized quadrature procedures discontinuous and singular fields can be regularized however regularization changes the algebraic structure of the solving equations and this can lead to high errors we show how to acquire accurate and consistent results when regularization is carried out a three dimensional analysis of a tensile butt joint is performed through a regularized extended finite element method the accuracy obtained via gaussian quadrature is compared with that obtained by means of cubpack adaptive quadrature fortran tool the use of regularized functions with non compact and compact support is investigated through an error evaluation procedure based on the use of their fourier transform the proposed procedure leads to the remarkable conclusion that regularized delta functions with non compact support exhibit superior performance copyright 2014 john wiley sons ltd

multilevel matrix multiplication and fast solution of integral equations abstract a fast multigrid approach is described for the task of calculating k x y u y dy for each x r d discretizing by an equidistant grid with n points and meshsize h and approximating the integrations to o h2s accuracy it is shown that the complexity of this calculation can be reduced from o n2 to o sn provided the kernel k is sufficiently smooth for potential type kernels the complexity is reduced to o sn log n corresponding integral equations can be solved to a similar accuracy in basically the same amount of work using a special kind of distributed relaxation in a multigrid algorithm one and two dimensional numerical tests and theoretical derivations of optimal strategies are reported the method is applicable to the task of multiplying by any matrix with appropriate smoothness properties including most types of many body interactions

a regularized xfem framework for embedded cohesive interfaces this work focuses on xfem modeling of elastodamaging interfaces characterized by regularized kinematics after regularization of the displacement discontinuity a process zone of finite width with continuous strain fields emerges the stress fields are independently energy deduced modeled via specific constitutive laws and subsequently introduced within the virtual work principle the methodology can address cohesive interfaces with vanishing and finite thickness in a unified way two dimensional results for a strain localization problem and several delamination tests are discussed in detail and compared with previous numerical experimental results