generalized fictitious methods for fluid structure interactions analysis and simulations we present a new fictitious pressure method for fluid structure interaction fsi problems in incompressible flow by generalizing the fictitious mass and damping methods we published previously in 1 the fictitious pressure method involves modification of the fluid solver whereas the fictitious mass and damping methods modify the structure solver we analyze all fictitious methods for simplified problems and obtain explicit expressions for the optimal reduction factor convergence rate index at the fsi interface 2 this analysis also demonstrates an apparent similarity of fictitious methods to the fsi approach based on robin boundary conditions which have been found to be very effective in fsi problems we implement all methods including the semi implicit robin based coupling method in the context of spectral element discretization which is more sensitive to temporal instabilities than low order methods however the methods we present here are simple and general and hence applicable to fsi based on any other spatial discretization in numerical tests we verify the selection of optimal values for the fictitious parameters for simplified problems and for vortex induced vibrations viv even at zero mass ratio for ever resonance we also develop an empirical a posteriori analysis for complex geometries and apply it to 3d patient specific flexible brain arteries with aneurysms for very large deformations we demonstrate that the fictitious pressure method enhances stability and convergence and is comparable or better in most cases to the robin approach or the other fictitious methods

asynchronous distance between homologous dna sequences the distance between homologous dna sequences of two species is proposed to be 1 4 ln det p where p is the conditional probability matrix specifying the proportions of the various nucleotides in the second sequence corresponding to each of the four nucleotides in the first sequence a probability model is described which supports this choice of distance distance measures based on a constant evolutionary rate assumption are described and compared with the proposed measure sampling properties of both types of distance are examined and we conclude by applying the distance measures to mitochondrial dna sequences of the hominoids

applying the hp d version of the fem to locally enhance dimensionally reduced models abstract a modification of the hp d method is presented which allows to locally enhance a dimensionally reduced finite element approximation with an ansatz based on a more accurate i e dimensionally higher model the procedure is based on a hierarchical domain decomposition which is applied to couple discretizations of different types of mechanical models like beams plates and solids thereby the error of the underlying mathematical model can be reduced in critical regions by introducing locally a physically more sophisticated model two examples are given to demonstrate the basic properties of the proposed method in the first example the solution of a one dimensional beam model is improved locally by a two dimensional finite element computation in the second example the local enhancement of a reissner mindlin plate by a fully three dimensional continuum approach is considered yielding a significant reduction of the model error

axisymmetric pressure boundary loading for finite deformation analysis using p fem follower loads i e loads which depend on the boundary displacements by definition frequently occur in finite deformation boundary value problems restricting to axisymmetrical applications we provide analytical and numerical solutions for a set of problems in compressible neo hookean materials so to serve as benchmark problems for verifying the accuracy and efficiency of various fe methods for follower load applications thereafter the weak formulation for the follower load in 3 d domain is reduced to an axisymmetrical setting and subsequently consistently linearized in the framework of p fems exploiting the blending function mapping techniques the set of axisymmetric benchmark solutions is compared to numerical experiments in which the results obtained by a p fem code are compared to these obtained by a state of the art commercial h fem code and to the exact results these demonstrate the efficiency and accuracy of p fems when applied to problems in finite deformations with follower loads 2006 elsevier b v all rights reserved

a semi local spectral hp element solver for linear elasticity problems summary we develop an efficient semi local method for speeding up the solution of linear systems arising in spectral hp element discretization of the linear elasticity equations the main idea is to approximate the element wise residual distribution with a localization operator we introduce in this paper and subsequently solve the local linear system additionally we decouple the three directions of displacement in the localization operator hence enabling the use of an efficient low energy preconditioner for the conjugate gradient solver this approach is effective for both nodal and modal bases in the spectral hp element method but here we focus on the modal hierarchical basis in numerical tests we verify that there is no loss of accuracy in the semi local method and we obtain good parallel scalability and substantial speed up compared to the original formulation in particular our tests include both structure only and fluid structure interaction problems with the latter modeling a 3d patient specific brain aneurysm copyright 2014 john wiley sons ltd

normaliz algorithms for affine monoids and rational cones abstract normaliz is a program for the computation of hilbert bases of rational cones and the normalizations of affine monoids it may also be used for solving diophantine linear systems in this paper we present the algorithms implemented in the program

construction of shape functions for the h and p versions of the fem using tensorial product this paper presents an uniform and unified approach to construct h and p shape functions for quadrilaterals triangles hexahedral and tetrahedral based on the tensorial product of one dimensional lagrange and jacobi polynomials the approach uses indices to denote the one dimensional polynomials in each tensorization direction the appropriate manipulation of the indices allows to obtain hierarchical or non hierarchical and inter element c continuous or non continuous bases for the one dimensional elements quadrilaterals triangles and hexahedral the optimal weights of the jacobi polynomials are determined the sparsity profiles of the local mass and stiffness matrices plotted and the condition numbers calculated a brief discussion of the use of sum factorization and computational implementation is considered

from electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex the electrostatic interpretation of the jacobi gauss quadrature points is exploited to obtain interpolation points suitable for approximation of smooth functions defined on a simplex moreover several new estimates based on extensive numerical studies for approximation along the line using jacobi gauss lobatto quadrature points as the nodal sets are presented the electrostatic analogy is extended to the two dimensional case with the emphasis being on nodal sets inside a triangle for which two very good matrices of nodal sets are presented the matrices are evaluated by computing the lebesgue constants and they share the property that the nodes along the edges of the simplex are the gauss lobatto quadrature points of the chebyshev and legendre polynomials respectively this makes the resulting nodal sets particularly well suited for integration with conventional spectral methods and supplies a new nodal basis for h p finite element methods

basis functions for triangular and quadrilateral high order elements in this paper we present a unified description of new spectral bases suitable for high order hp finite element discretizations on hybrid two dimensional meshes consisting of triangles and quadrilaterals all bases presented are for c0 continuous discretizations and are described both as modal and as mixed modal nodal expansions general jacobi polynomials of mixed weights are employed that accommodate automatic exact numerical quadratures generalized tensor products and variable expansion order in each element the approximation properties of the bases are analyzed in the context of the projection linear advection and diffusion operators

low energy basis preconditioning for elliptic substructured solvers based on unstructured spectral hp element discretization abstract the development and application of three dimensional unstructured hierarchical spectral hp element algorithms has highlighted the need for efficient preconditioning for elliptic solvers building on the work of bica ph d thesis courant institute new york university 1997 we have developed an efficient preconditioning strategy for substructured solvers based on a transformation of the expansion basis to a low energy basis in this numerically derived basis the strong coupling between expansion modes in the original basis is reduced thus making it amenable to block diagonal preconditioning the efficiency of the algorithm is maintained by developing the new basis on a symmetric reference element and ignoring in the preconditioning step the role of the jacobian of the mapping from the reference to the global element by applying an additive schwarz block preconditioner to the low energy basis combined with a coarse space linear vertex solver we have observed reductions in execution time of up to three times for tetrahedral elements and 10 times for prismatic elements when compared to a standard diagonal preconditioner full details of the implementation and validation of the tetrahedral and prismatic element preconditioning strategy are set out below

bdf like methods for nonlinear dynamic analysis we present several time integration algorithms of second order accuracy that are numerically simple and effective for nonlinear elastodynamic problems these algorithms are based on a general four step scheme that has a resemblance to the backward differentiation formulas we also present an extension to the composite strategy of the bathe method appropriate values for the algorithmic parameters are determined based on considerations of stability and dissipativity and less dissipative members of each algorithm have been identified we demonstrate the convergence characteristics of the proposed algorithms with a nonlinear dynamic problem having analytic solutions and test these algorithms with several three dimensional nonlinear elastodynamic problems involving large deformations and rotations employing st venant kirchhoff and compressible neo hookean hyperelastic material models these tests show that stable computations are obtained with the proposed algorithms in nonlinear situations where the trapezoidal rule encounters a well known instability

an algorithm for computing fekete points in the triangle on the line and its tensor products fekete points are known to be the gauss lobatto quadrature points but unlike high order quadrature fekete points generalize to non tensor product domains such as the triangle thus fekete points might serve as an alternative to the gauss lobatto points for certain applications in this work we present a new algorithm to compute fekete points and give results up to degree 19 for the triangle for degree d 10 these points have the smallest lebesgue constant currently known the computations validate a conjecture of bos j approx theory 64 1991 pp 271 280 that fekete points along the boundary of the triangle are the one dimensional gauss lobatto points

fully tensorial nodal and modal shape functions for triangles and tetrahedra this paper presents nodal and modal shape functions for triangle and tetrahedron finite elements the functions are constructed based on the fully tensorial expansions of one dimensional polynomials expressed in barycentric co ordinates the nodal functions obtained from the application of the tensorial procedure are the standard h lagrange shape functions presented in the literature the modal shape functions use jacobi polynomials and have a natural global c0 inter element continuity an efficient gauss jacobi numerical integration procedure is also presented to decrease the number of points for the consistent integration of the element matrices an example illustrates the approximation properties of the modal functions copyright 2005 john wiley sons ltd

toward a universal h p adaptive finite element strategy part 1 constrained approximation and data structure abstract this is the first of a series of papers devoted to the development of an adaptive finite element method to solve various boundary value problems arising in solid and fluid mechanics the paper deals with the general concepts of h p finite element methods constrained approximation and data structure and related issues several numerical examples illustrate the main ideas of the method

a triangular spectral element method applications to the incompressible navier stokes equations abstract encouraged by the success of spectral elements methods in computational fluid dynamics and p type finite element methods in structural mechanics we wish to extend these ideas to solving high order polynomial approximations on triangular domains as the next generation of spectral element solvers we introduce here a complete formulation using a modal basis which has been implemented in a new code ne t r the new basis has the following properties jacobi polynomials of mixed weights semi orthogonality hierarchical structure generalized tensor warped product variable order and a new apex co ordinate system allowing automated integration with gaussian quadrature we have discussed the formulation using a matrix notation which allows for an easy interpretation of the forward and backward transformations we use this notation to formulate the linear advection and helmholtz equations in an efficient manner and show that we recover the following properties well conditioned matrices asymptotic operation count of o n 3 scaling o n 2 of the spectral radius of the weak convective operator and exponential convergence using polynomials up to 40 having constructed and numerically analysed these equations we are then able to solve the incompressible navier stokes equations using a high order splitting scheme we demonstrate a variety of results showing exponential convergence using both deformed and straight triangular subdomains for both the stokes and navier stokes problems

an eigen based high order expansion basis for structured spectral elements we present an eigen based high order expansion basis for the spectral element approach with structured elements the new basis exhibits a numerical efficiency significantly superior in terms of the conditioning of coefficient matrices and the number of iterations to convergence for the conjugate gradient solver to the commonly used jacobi polynomial based expansion basis this basis results in extremely sparse mass matrices and it is very amenable to the diagonal preconditioning ample numerical experiments demonstrate that with the new basis and a simple diagonal preconditioner the number of conjugate gradient iterations to convergence has essentially no dependence or only a very weak dependence on the element order the expansion bases are constructed by a tensor product of a set of special one dimensional 1d basis functions the 1d interior modes are constructed such that the interior mass and stiffness matrices are simultaneously diagonal and have identical condition numbers the 1d vertex modes are constructed to be orthogonal to all the interior modes the performance of the new basis has been investigated and compared with other expansion bases

high order splitting methods for the incompressible navier stokes equations abstract a new pressure formulation for splitting methods is developed that results in high order time accurate schemes for the solution of the incompressible navier stokes equations in particular improved pressure boundary conditions of high order in time are introduced that minimize the effect of erroneous numerical boundary layers induced by splitting methods a new family of stiffly stable schemes is employed in mixed explicit implicit time intgration rules these schemes exhibit much broader stability regions as compared to adams family schemes typically used in splitting methods their stability properties remain almost constant as the accuracy of the integration increases so that robust third or higher order time accurate schemes can readily be constructed that remain stable at relatively large cfl number the new schemes are implemented within the framework of spectral element discretizations in space so that flexibility and accuracy is guaranteed in the numerical experimentation a model stokes problem is studied in detail and several examples of navier stokes solutions of flows in complex geometries are reported comparison is made with the previously used first order in time spectral element splitting and non splitting e g uzawa schemes high order splitting spectral element methods combine accuracy in space and time and flexibility in geometry and thus can be very efficient in direct simulations of turbulent flows in complex geometries

adaptive hpq finite element methods of hierarchical models for plate and shell like structures abstract a four step adaptive strategy for hpq finite element approximations of hierarchical models of the thin plate and shell like structures is derived based on a priori and a posteriori error estimations this strategy controls three parameters the model level q the mesh size h and the approximation order p so as to produce quasi optimal hierarchical models and finite element meshes numerical results are given supporting the theoretical results

a convergence study of a new partitioned fluid structure interaction algorithm based on fictitious mass and damping we develop analyze and validate a new method for simulating fluid structure interactions fsis which is based on fictitious mass and fictitious damping in the structure equation we employ a partitioned method for the fluid and structure motions in conjunction with sub iteration and aitken relaxation in particular the use of such fictitious parameters requires sub iterations in order to reduce the induced error in addition to the local temporal truncation error to this end proper levels of tolerance for terminating the sub iteration procedure have been obtained in order to recover the formal order of temporal accuracy for the coupled fsi problem these fictitious terms have a significant effect leading to better convergence rate and hence substantially smaller number of sub iterations through analysis we identify the proper range of these parameters which we then verify by corresponding numerical tests we implement the method in the context of spectral element discretization which is more sensitive than low order methods to numerical instabilities arising in the explicit fsi coupling however the method we present here is simple and general and hence applicable to fsi based on any other discretization we demonstrate the effectiveness of the method in applications involving 2d vortex induced vibrations viv and in 3d flexible arteries with structural density close to blood density we also present 3d results for a patient specific aneurysmal flow under pulsatile flow conditions examining in particular the sensitivity of the results on different values of the fictitious parameters

analytical and computational assessment of locking in the hp finite element method locking is the phenomenon by which the numerical approximation of parameter dependent problems deteriorates for values of the parameter close to a limiting value in this paper we give a definition of locking and develop precise computable and analytic ways to quantify it using the example of nearly incompressible elasticity we show by means of computational and theoretical results the difference between the h version and php version in combatting locking our results establish the superiority of high order elements both h p and hp when the standard variational form is used we also discuss other issues such as curved elements mixed methods and locking phenomena for problems over anisotropic materials and over thin domains

mixed spectral hp element formulation for nonlinear elasticity we present a mixed formulation for high order spectral hp element methods and investigate its stability in large strain analysis of nearly incompressible solids specifically we employ the displacement pressure formulation and use a pressure projection method along with general jacobi polynomial bases to derive the discrete equations we first study the convergence of the method for smooth and singular solutions of elastic and hyperelastic problems for various combinations of mixed elements subsequently we investigate numerically the validity of ellipticity and inf sup conditions for various test problems including cases with large deformation we find that the inf sup condition is satisfied even when the pressure interpolation order op is equal to the displacement order ou but to resolve the locking free behavior it is required that op ou 1 and this choice also gives the best convergence rate in large deformation analysis surprisingly the results are more stable for ou op ou k because the ellipticity in kernel is better satisfied with higher k as ou increases compared to the displacement only formulation the mixed formulation seems to have superior accuracy and it is overall more efficient especially for problems with high bulk modulus

fluid structure interaction problems with strong added mass effect in this paper the so called added mass effect is investigated from a different point of view of previous publications the monolithic fluid structure problem is partitioned using a static condensation of the velocity terms following this procedure the classical stabilized projection method for incompressible fluid flows is introduced the procedure allows obtaining a new pressure segregated scheme for fluid structure interaction problems which has good convergent characteristics even for biomechanical application where the added mass effect is strong the procedure reveals its power when it is shown that the same projection technique must be implemented in staggered fsi methods copyright 2009 john wiley sons ltd

higher order triangular basis functions and solution performance of the cg method new sets of hierarchical higher order basis functions in fem for triangle elements are constructed using a systematic orthogonalization approach that yield better conditioning and investigated with different preconditioners jacobi icc ilu and sainv presented theoretical and numerical results indicate that certain preconditioners are insensitive to the condition number of the basis functions advantageous properties of basis functions in relation to preconditioners are viewed from the perspective of similarity and congruence transforms of the basis functions this provides an alternative view for a systematic construction of the basis functions that couples with a specific preconditioner to optimize the solution performance

a spectral element method for fluid dynamics laminar flow in a channel expansion a spectral element method that combines the generality of the finite element method with the accuracy of spectral techniques is proposed for the numerical solution of the incompressible navier stokes equations in the spectral element discretization the computational domain is broken into a series of elements and the velocity in each element is represented as a high order lagrangian interpolant through chebyshev collocation points the hyperbolic piece of the governing equations is then treated with an explicit collocation scheme while the pressure and viscous contributions are treated implicitly with a projection operator derived from a variational principle the implementation of the technique is demonstrated on a one dimensional inflow outflow advection diffusion equation and the method is then applied to laminar two dimensional separated flow in a channel expansion comparisons are made with experiment and previous numerical work

a general high order finite element formulation for shells at large strains and finite rotations for hyperelastic shells with finite rotations and large strains a p finite element formulation is presented accommodating general kinematic assumptions interpolation polynomials and particularly general three dimensional hyperelastic constitutive laws this goal is achieved by hierarchical high order shell models the tangent stiffness matrices for the hierarchical shell models are derived by computer algebra both non hierarchical nodal as well as hierarchical element shape functions are admissible numerical experiments show the high order formulation to be less prone to locking effects copyright 2003 john wiley sons ltd

a p version finite element approach for two and three dimensional problems of the j2 flow theory with non linear isotropic hardening in this paper an implementation of a two and three dimensional p version approach to the j 2 flow theory with non linear isotropic hardening for small displacements and small strains is presented based on higher order quadrilateral and hexahedral element formulations a newton raphson iteration scheme combined with a radial return algorithm is applied to find approximate solutions for the underlying physically non linear model problem curved boundaries are taken care of with the blending function method allowing an accurate representation of geometry with only a few p elements numerical examples demonstrate that the p version supplies efficient and accurate approximations to this class of physically non linear problems