the duality principle in fuzzy set theory abstract a rigorous mathematical approach to extending nonfuzzy structures to fuzzy sets is presented this approach is based on classical ideas of duality in analysis and inverse image in set theory we apply these new techniques to characterizing indistinguishability relations fuzzy interval orders and fuzzy operations zadeh s extension principle is characterized in terms of the duality principle

a comparison of high order polynomial and wave based methods for helmholtz problems the application of computational modelling to wave propagation problems is hindered by the dispersion error introduced by the discretisation two common strategies to address this issue are to use high order polynomial shape functions e g hp fem or to use physics based or trefftz methods where the shape functions are local solutions of the problem typically plane waves both strategies have been actively developed over the past decades and both have demonstrated their benefits compared to conventional finite element methods but they have yet to be compared in this paper a high order polynomial method p fem with lobatto polynomials and the wave based discontinuous galerkin method are compared for two dimensional helmholtz problems a number of different benchmark problems are used to perform a detailed and systematic assessment of the relative merits of these two methods in terms of interpolation properties performance and conditioning it is generally assumed that a wave based method naturally provides better accuracy compared to polynomial methods since the plane waves or bessel functions used in these methods are exact solutions of the helmholtz equation results indicate that this expectation does not necessarily translate into a clear benefit and that the differences in performance accuracy and conditioning are more nuanced than generally assumed the high order polynomial method can in fact deliver comparable and in some cases superior performance compared to the wave based dgm in addition to benchmarking the intrinsic computational performance of these methods a number of practical issues associated with realistic applications are also discussed

a stable penalty method for the compressible navier stokes equations iii multidimensional domain decomposition schemes this paper concluding the trilogy develops schemes for the stable solution of wave dominated unsteady problems in general three dimensional domains the schemes utilize a spectral approximation in each subdomain and asymptotic stability of the semidiscrete schemes is established the complex computational domains are constructed by using nonoverlapping quadrilaterals in the two dimensional case and hexahedrals in the three dimensional space to illustrate the ideas underlying the multidomain method a stable scheme for the solution of the three dimensional linear advection diffusion equation in general curvilinear coordinates is developed the analysis suggests a novel yet simple stable treatment of geometric singularities like edges and vertices the theoretical results are supported by a two dimensional implementation of the scheme the main part of the paper is devoted to the development of a spectral multidomain scheme for the compressible navier stokes equations on conservation form and a unified approach for dealing with the open boundaries and subdomain boundaries is presented well posedness and asymptotic stability of the semidiscrete scheme is established in a general curvilinear volume with special attention given to a hexahedral domain the treatment includes a stable procedure for dealing with boundary conditions at a solid wall the efficacy of the scheme for the compressible navier stokes equations is illustrated by obtaining solutions to subsonic and supersonic boundary layer flows with various types of boundary conditions the results are found to agree with the solution of the compressible boundary layer equations

optimal decomposition of the domain in spectral methods for wave like phenomena a strategy for determining the optimal number of grid points and subdomains in a spectral method with domain decomposition on a serial computer is presented the rapidly growing computational cost for large numbers of grid points in each subdomain is balanced against the exponential convergence for spectral approximation of smooth functions and the optimum is found as the number of grid points and subdomains that gives the minimal computational cost for a given accuracy the typical length scale of the problem is found to influence the number of subdomains but not the number of grid points within each subdomain

nodal high order methods on unstructured grids we present a convergent high order accurate scheme for the solution of linear conservation laws in geometrically complex domains as our main example we include a detailed development and analysis of a scheme for the time domain solution of maxwell s equations in a three dimensional domain the fully unstructured spatial discretization is made possible by the use of a high order nodal basis employing multivariate lagrange polynomials defined on the triangles and tetrahedra while the equations themselves are satisfied in a discontinuous galerkin form with the boundary conditions being enforced weakly through a penalty term accuracy stability and convergence of the semidiscrete approximation to maxwell s equations is established rigorously and bounds on the growth of the global divergence error are provided concerns related to efficient implementations are discussed in detail this sets the stage for the presentation of examples verifying the theoretical results and illustrating the versatility flexibility and robustness when solving two and three dimensional benchmark problems in computational electromagnetics pure scattering as well as penetration is discussed and high parallel performance of the scheme is demonstrated

from h to p efficiently implementing finite and spectral hp element methods to achieve optimal performance for low and high order discretisations the spectral hp element method can be considered as bridging the gap between the traditionally low order finite element method on one side and spectral methods on the other side consequently a major challenge which arises in implementing the spectral hp element methods is to design algorithms that perform efficiently for both low and high order spectral hp discretisations as well as discretisations in the intermediate regime in this paper we explain how the judicious use of different implementation strategies can be employed to achieve high efficiency across a wide range of polynomial orders furthermore based upon this efficient implementation we analyse which spectral hp discretisation which specific combination of mesh size h and polynomial order p minimises the computational cost to solve an elliptic problem up to a predefined level of accuracy we investigate this question for a set of both smooth and non smooth problems

fast formation and assembly of finite element matrices with application to isogeometric linear elasticity abstract recently a new formation and assembly strategy was proposed in 1 which resulted in significant speedups in the formation and assembly time of the galerkin mass matrix in isogeometric analysis the strategy relies on three key ingredients 1 assembly row by row instead of element by element and an efficient formation strategy based on 2 sum factorization and 3 weighted quadrature that is applied to each specific row of the matrix compared to traditional element procedures applied to three dimensional problems the computational complexity is lowered from o p 9 per degree of freedom to o p 4 where p is the order of polynomials this is close to the theoretical minimum of o p 3 attained by for example collocation consequently this type of formation and assembly scales favorably with polynomial degree which opens the way for high order isogeometric analysis employing k refinement that is use of maximally smooth higher order splines in this work we discuss various important details for the practical implementation of the weighted quadrature formation strategy proposed in 1 specifically we extend the weighted quadrature scheme to accurately integrate the elements of the stiffness matrix in linear elasticity and propose a means of distributing quadrature points for non uniform mixed continuity spline spaces furthermore we discuss efficient access and assignment into the prevalent sparse matrix data structures namely compressed sparse row csr and compressed sparse column csc in particular row by row or column by column assembly allows matrix rows or columns respectively to be formed contiguously in the storage order of the sparse matrix thereby minimizing the memory overhead and eliminating the addition assignment operation on sparse matrices several three dimensional benchmark problems illustrate the efficiency and efficacy of the proposed formation and assembly technique applied to isogeometric linear elasticity we show that the accuracy of full gauss quadrature is maintained while the computational burden of forming the matrix equations is significantly reduced

robust untangling of curvilinear meshes this paper presents a technique that allows to untangle high order curvilinear meshes the technique makes use of unconstrained optimization where element jacobians are constrained to lie in a prescribed range through moving log barriers the untangling procedure starts from a possibly invalid curvilinear mesh and moves mesh vertices with the objective of producing elements that all have bounded jacobians bounds on jacobians are computed using the results of johnen et al 2012 2013 1 2 the technique is applicable to any kind of polynomial element for surface volume hybrid or boundary layer meshes a series of examples demonstrate both the robustness and the efficiency of the technique the final example involving a time explicit computation shows that it is possible to control the stable time step of the computation for curvilinear meshes through an alternative element deformation measure

extending fuzzy relations abstract in this work a way to extend in very general conditions the action of fuzzy relations in a referential x to relations in f x is studied and we consider the lattice structure and other properties of these extended relations

a high order discontinuous galerkin fourier incompressible 3d navier stokes solver with rotating sliding meshes we present the development of a sliding mesh capability for an unsteady high order order 3 h p discontinuous galerkin solver for the three dimensional incompressible navier stokes equations a high order sliding mesh method is developed and implemented for flow simulation with relative rotational motion of an inner mesh with respect to an outer static mesh through the use of curved boundary elements and mixed triangular quadrilateral meshes a second order stiffly stable method is used to discretise in time the arbitrary lagrangian eulerian form of the incompressible navier stokes equations spatial discretisation is provided by the symmetric interior penalty galerkin formulation with modal basis functions in the x y plane allowing hanging nodes and sliding meshes without the requirement to use mortar type techniques spatial discretisation in the z direction is provided by a purely spectral method that uses fourier series and allows computation of spanwise periodic three dimensional flows the developed solver is shown to provide high order solutions second order in time convergence rates and spectral convergence when solving the incompressible navier stokes equations on meshes where fixed and rotating elements coexist in addition an exact implementation of the no slip boundary condition is included for curved edges circular arcs and naca 4 digit airfoils where analytic expressions for the geometry are used to compute the required metrics the solver capabilities are tested for a number of two dimensional problems governed by the incompressible navier stokes equations on static and rotating meshes the taylor vortex problem a static and rotating symmetric naca0015 airfoil and flows through three bladed cross flow turbines in addition three dimensional flow solutions are demonstrated for a three bladed cross flow turbine and a circular cylinder shadowed by a pitching naca0012 airfoil

a characterization of the extension principle abstract we provide an alternative characterization of the extension principle of fuzzy sets this characterization is based upon using relations to represent mappings we apply this new characterization to develop an extension for non deterministic mappings

pyramid algorithms for bernstein bezier finite elements of high nonuniform order in any dimension the archetypal pyramid algorithm is the de casteljau algorithm which is a standard tool for the evaluation of bezier curves and surfaces pyramid algorithms replace an operation on a single high order polynomial by a recursive sequence of self similar affine combinations and are ubiquitous in computer aided geometric design for computations involving high order curves and surfaces pyramid algorithms have received no attention whatsoever from the high or low order finite element community we develop and analyze pyramid algorithms for the efficient handling of all of the basic finite element building blocks including the assembly of the element load vectors and element stiffness matrices the complexity of the algorithm for generating the element stiffness matrix is optimal a new nonuniform order variant of the de casteljau algorithm is developed that is applicable to the variable polynomial order case but incurs no additional complexity compared with the original algorithm the work provides the

a distortion measure to validate and generate curved high order meshes on cad surfaces with independence of parameterization the first author was sponsored by cur from diue of the generalitat de catalunya and the european social fund grants fi dgr 11 and be dgr 12 and by the ferran sunyer i balaguer foundation grant fsb 13 the work of the corresponding author was partially supported by the boeing co us air force office of scientific research european commission through the boeing mit alliance the computational math program marie sklodowska curie actions hipermegaflows individual fellowship grant

a collocated c0 finite element method reduced quadrature perspective cost comparison with standard finite elements and explicit structural dynamics summary we demonstrate the potential of collocation methods for efficient higher order analysis on standard nodal finite element meshes we focus on a collocation method that is variationally consistent and geometrically flexible converges optimally embraces concepts of reduced quadrature and leads to symmetric stiffness and diagonal consistent mass matrices at the same time it minimizes the evaluation cost per quadrature point thus reducing formation and assembly effort significantly with respect to standard galerkin finite element methods we provide a detailed review of all components of the technology in the context of elastodynamics that is weighted residual formulation nodal basis functions on gauss lobatto quadrature points and symmetrization by averaging with the ultra weak formulation we quantify potential gains by comparing the computational efficiency of collocated and standard finite elements in terms of basic operation counts and timings our results show that collocation is significantly less expensive for problems dominated by the formation and assembly effort such as higher order elastostatic analysis furthermore we illustrate the potential of collocation for efficient higher order explicit dynamics throughout this work we advocate a straightforward implementation based on simple modifications of standard finite element codes we also point out the close connection to spectral element methods where many of the key ideas are already established copyright 2014 john wiley sons ltd

hp submeshing via non conforming finite element methods abstract non conformity in the hp version can involve incompatibility in both the degrees and the meshes between adjoining subdomains in this paper we show how the mortar finite element method m0 and two new variants m1 m2 can be used to join together such incompatible hp sub discretizations our results show optimality of the resulting non conforming method for various h p and hp discretizations including the case of exponential hp convergence over geometric meshes we also present numerical results for the lagrange multiplier when the method is implemented via a mixed method three dimensional considerations suggest that our methods m1 m2 are easier to generalize to arbitrary meshes than m0

fuzzy relations on fuzzy and non fuzzy numbers fast computation formulas abstract fast computation formulas for the operation supx ymin h x z y x y are proved in case the membership functions h z of fuzzy numbers and non fuzzy numbers are triangular rectangular and the function of dirac

an accelerated central cutting plane algorithm for linear semi infinite programming an algorithm for linear semi infinite programming is presented which accelerates the convergence of the central cutting plane algorithm first proposed in 4 compared with other algorithms the algorithm in 4 has the advantage of being applicable under mild conditions and of providing feasible solutions however its convergence has been shown to be rather slow in practical instances the algorithm proposed in this paper introduces a simple acceleration scheme which gives faster convergence as confirmed by several examples as well as an interval of prefixed length containing the optimum value it is also shown that the algorithm provides a solution of the dual problem and that it can be used for convex semi infinite programming too

high order discontinuous galerkin schemes on general 2d manifolds applied to the shallow water equations an innovating approach is proposed to solve vectorial conservation laws on curved manifolds using the discontinuous galerkin method this new approach combines the advantages of the usual approaches described in the literature the vectorial fields are expressed in a unit non orthogonal local tangent basis derived from the polynomial mapping of curvilinear triangle elements while the convective flux functions are written is the usual 3d cartesian coordinate system the number of vectorial components is therefore minimum and the tangency constraint is naturally ensured while the method remains robust and general since not relying on a particular parametrization of the manifold the discontinuous galerkin method is particularly well suited for this approach since there is no continuity requirement between elements for the tangent basis definition the possible discontinuities of this basis are then taken into account in the riemann solver on inter element interfaces the approach is validated on the sphere using the shallow water equations for computing standard atmospheric benchmarks in particular the williamson test cases are used to analyze the impact of the geometry on the convergence rates for discretization error the propagation of gravity waves is eventually computed on non conventional irregular curved manifolds to illustrate the robustness and generality of the method

algorithmic estimate for aggregated powerset relations abstract this work studies the level cut representation of those fuzzy powerset relations obtained by aggregation this representation is used to present algorithmic methods of calculation of values for the extended relations connections with zadeh s principle of extension are considered

tetrahedralhpfinite elements we present a new discretisation for the incompressible navier stokes equations that extends spectral methods to three dimensional complex domains consisting of tetrahedral subdomains the algorithm is based on standard concepts ofhpfinite elements as well as tensorial spectral elements this new formulation employs a hierarchical modal basis constructed from a new apex co ordinate system which retains a generalised tensor product these properties enable the development of computationally efficienct algorithms for use on standard finite volume unstructured meshes a detailed analysis is presented that documents the stability and exponential convergence of the method and several flow cases are simulated and compared with analytical and experimental results

the p and h p versions of the finite element method basic principles and properties in the classical form of the finite element method called the hversion piecewise polynomials of fixed degree p are used and the mesh size h is decreased for accuracy in this paper we discuss the fundamental theoretical ideas behind the relatively recent p version and h p version in the p version a fixed mesh is used and p is allowed to increase the h p version combines both approaches the authors describe and explain the basic properties and characteristics of these newer versions especially in areas where their behavior is significantly different from that of the h version simplified proofs of key concepts are included and computational illustrations of several results are provided a benchmark comparison between the various versions in included

a compatible and conservative spectral element method on unstructured grids we derive a formulation of the spectral element method which is compatible on very general unstructured three dimensional grids here compatible means that the method retains discrete analogs of several key properties of the divergence gradient and curl operators the divergence and gradient are anti adjoints the negative transpose of each other the curl is self adjoint and annihilates the gradient operator and the divergence annihilates the curl the adjoint relations hold globally and at the element level with the inclusion of a natural discrete element boundary flux term we then discretize the shallow water equations on the sphere using the cubed sphere grid and show that compatibility allows us to locally conserve mass energy and potential vorticity conservation is obtained without requiring the equations to be in conservation form the conservation is exact assuming exact time integration

a spectral finite element approach to modeling soft solids excited with high frequency harmonic loads abstract an approach for efficient and accurate finite element analysis of harmonically excited soft solids using high order spectral finite elements is presented and evaluated the helmholtz type equations used to model such systems suffer from additional numerical error known as pollution when excitation frequency becomes high relative to stiffness i e high wave number which is the case for example for soft tissues subject to ultrasound excitations the use of high order polynomial elements allows for a reduction in this pollution error but requires additional consideration to counteract runge s phenomenon and or poor linear system conditioning which has led to the use of spectral element approaches this work examines in detail the computational benefits and practical applicability of high order spectral elements for such problems the spectral elements examined are tensor product elements i e quad or brick elements of high order lagrangian polynomials with non uniformly distributed gauss lobatto legendre nodal points a shear plane wave example is presented to show the dependence of the accuracy and computational expense of high order elements on wave number then a convergence study for a viscoelastic acoustic structure interaction finite element model of an actual ultrasound driven vibroacoustic experiment is shown the number of degrees of freedom required for a given accuracy level was found to consistently decrease with increasing element order however the computationally optimal element order was found to strongly depend on the wave number