2 cocycles on the algebra of differential operators soit c t t 1 l anneau des polynomes de laurent sur un corps complexe c soit a c t t 1 d l algebre des operateurs differentiels sur c t t 1 soit 7b u l algebre des operateurs differentiels 7b u a gl u c sur c t t 1 gl u c tout 2 cocycle sur a est equivalent a un multiple de tout 2 cocycle sur 7b u est equivalent a un multiple de

a hierarchical approach to adaptive local refinement in isogeometric analysis abstract adaptive local refinement is one of the key issues in isogeometric analysis in this article we present an adaptive local refinement technique for isogeometric analysis based on extensions of hierarchical b splines we investigate the theoretical properties of the spline space to ensure fundamental properties like linear independence and partition of unity furthermore we use concepts well established in finite element analysis to fully integrate hierarchical spline spaces into the isogeometric setting this also allows us to access a posteriori error estimation techniques numerical results for several different examples are given and they turn out to be very promising

isogeometric analysis and hierarchical refinement for higher order phase field models abstract while the interest in higher order models in physics and mechanics grows their numerical simulation still poses a challenge especially for arbitrary shaped three dimensional domains this contribution presents the mathematical framework as well as the application to different problems in the field of material science fracture mechanics and diffusion problems all models under consideration require at least c 1 continuity which prevents the application of standard finite element analysis and local mesh refinements introducing isogeometric analysis iga for the discretization in a finite element framework enables us to deal with these requirements moreover a general hierarchical refinement scheme based on a subdivision projection is presented here for one two and three dimensional simulations this technique allows to enhance the approximation space using finer splines on each level but preserves the partition of unity as well as the continuity properties of the original discretization using this mathematical framework the improved convergence of a kuramoto sivashinsky model a mesh adapted thermal diffusion simulation and computations of a priori unknown crack propagation in different fracture modes underline the versatility of the presented hierarchical refinement scheme

a primal dual active set method and predictor corrector mesh adaptivity for computing fracture propagation using a phase field approach abstract in this paper we consider phase field based fracture propagation in elastic media the main purpose is the development of a robust and efficient numerical scheme to enforce crack irreversibility as a constraint we use a primal dual active set strategy which can be identified as a semi smooth newton method the active set iteration is merged with the newton iteration for solving the fully coupled nonlinear partial differential equation discretized using finite elements resulting in a single rapidly converging nonlinear scheme it is well known that phase field models require fine meshes to accurately capture the propagation dynamics of the crack because traditional estimators based on adaptive mesh refinement schemes are not appropriate we develop a predictor corrector scheme for local mesh adaptivity to reduce the computational cost this method is both robust and efficient and allows us to treat temporal and spatial refinements and to study the influence of model regularization parameters finally our proposed approach is substantiated with different numerical tests for crack propagation in elastic media and pressurized fracture propagation in homogeneous and heterogeneous media

spin and wedge representations of infinite dimensional lie algebras and groups abstract we suggest a purely algebraic construction of the spin representation of an infinite dimensional orthogonal lie algebra sections 1 and 2 and a corresponding group section 4 from this we deduce a construction of all level one highest weight representations of orthogonal affine lie algebras in terms of creation and annihilation operators on an infinite dimensional grassmann algebra section 3 we also give a similar construction of the level one representations of the general linear affine lie algebra in an infinite dimensional wedge space along these lines we construct the corresponding representations of the universal central extension of the group sln k t t 1 in spaces of sections of line bundles over infinite dimensional homogeneous spaces section 5

vertex operator representation of some quantum tori lie algebras we are defining the trigonometric lie subalgebras in bar x infty bar a infty bar b infty bar c infty bar d infty which are the natural generalization of the well known sin lie algebra the embedding formulas into bar x infty are introduced these algebras can be considered as some lie algebras of quantum tori an irreducible representation ofa b series of trigonometric lie algebras is constructed special cases of the trigonometric lie factor algebras which can be considered as a quantum preserving lie algebra structure deformation of the kac moody algebras are considered

representations of a symplectic type subalgebra of w n in this paper we classify the irreducible quasifinite highest weight modules over the symplectic type lie subalgebra of the lie algebra of all regular differential operators on circle that kill constants we also realize them in terms of the representations theory of the complex lie algebra gl m of infinite matrices with a finite number of non zero diagonals with entries in the algebra of truncated polynomials and the corresponding subalgebras of type c

characterizations of mixed binary convex quadratic representable sets representability results play a fundamental role in optimization since they provide characterizations of the feasible sets that arise from optimization problems in this paper we study the sets that appear in the feasibility version of mixed binary convex quadratic optimization problems we provide a complete characterization of the sets that can be obtained as the projection of such feasible regions in order to obtain this result we first provide a complete characterization of these sets in the special cases where 1 the feasible region is bounded 2 only binary extended variables are present and 3 only continuous variables are present

thermodynamically consistent phase field models of fracture variational principles and multi field fe implementations the computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies this can be overcome by a diffusive crack modeling based on the introduction of a crack phase field in this paper we outline a thermodynamically consistent framework for phase field models of crack propagation in elastic solids develop incremental variational principles and consider their numerical implementations by multi field finite element methods we start our investigation with an intuitive and descriptive derivation of a regularized crack surface functional that converges for vanishing length scale parameter to a sharp crack topology functional this functional provides the basis for the definition of suitable convex dissipation functions that govern the evolution of the crack phase field here we propose alternative rate independent and viscous over force models that ensure the local growth of the phase field next we define an energy storage function whose positive tensile part degrades with increasing phase field with these constitutive functionals at hand we derive the coupled balances of quasi static stress equilibrium and gradient type phase field evolution in the solid from the argument of virtual power here we consider a canonical two field setting for rate independent response and a time regularized three field formulation with viscous over force response it is then shown that these balances follow as the euler equations of incremental variational principles that govern the multi field problems these principles make the proposed formulation extremely compact and provide a perfect base for the finite element implementation including features such as the symmetry of the monolithic tangent matrices we demonstrate the performance of the proposed phase field formulations of fracture by means of representative numerical examples copyright 2010 john wiley sons ltd

a robust algorithm for finding the eigenvalues and eigenvectors of 3 3 symmetric matrices abstract many concepts in continuum mechanics are most easily understood in principal coordinates using these concepts in a numerical analysis requires a robust algorithm for finding the eigenvalues and eigenvectors of 3 3 symmetric matrices a robust algorithm for solving this eigenvalue problem is presented along with an analysis of the algorithm the special case of two or three nearly identical eigenvalues is examined in detail using an asymptotic analysis numerical results are shown that compare this algorithm with existing methods found in the literature the behavior of this algorithm is shown to be more reliable than the other methods with a minimal computational cost

mixed integer quadratic programming is in np mixed integer quadratic programming is the problem of optimizing a quadratic function over points in a polyhedral set where some of the components are restricted to be integral in this paper we prove that the decision version of mixed integer quadratic programming is in np thereby showing that it is np complete this is established by showing that if the decision version of mixed integer quadratic programming is feasible then there exists a solution of polynomial size this result generalizes and unifies classical results that quadratic programming is in np vavasis in inf process lett 36 2 73 77 17 and integer linear programming is in np borosh and treybig in proc am math soc 55 299 304 1 von zur gathen and sieveking in proc am math soc 72 155 158 18 kannan and monma in lecture notes in economics and mathematical systems vol 157 pp 161 172 springer 9 papadimitriou in j assoc comput mach 28 765 768 15

an isogeometric design through analysis methodology based on adaptive hierarchical refinement of nurbs immersed boundary methods and t spline cad surfaces we explore hierarchical refinement of nurbs as a basis for adaptive isogeometric and immersed boundary analysis we use the principle of b spline subdivision to derive a local refinement procedure which combines full analysis suitability of the basis with straightforward implementation in tree data structures and simple generalization to higher dimensions we test hierarchical refinement of nurbs for some elementary fluid and structural analysis problems in two and three dimensions and attain good results in all cases using the b spline version of the finite cell method we illustrate the potential of immersed boundary methods as a seamless isogeometric design through analysis procedure for complex engineering parts defined by t spline cad surfaces specifically a ship propeller and an automobile wheel we show that hierarchical refinement considerably increases the flexibility of this approach by adaptively resolving local features

comparison of high order finite element and discontinuous galerkin methods for phase field equations application to structural damage abstract phase field equations are used to model a wide range of multiphase problems such as separation of fluids solidification viscous fingering fracture and fatigue a wide variety of methods to numerically solve phase field equations can be found in the literature in particular high order methods are an effective option when accuracy improvement is desired in the first part of this work we analyze the accuracy and computational efficiency of the high order finite element method fem and discontinuous galerkin dg method applied to the second order allen cahn ac and fourth order cahn hilliard ch equations several schemes for time integration are used for these equations the explicit schemes are the forward euler classical fourth order runge kutta rk4 and the strong stability preserving ten stages fourth order runge kutta rkssp 10 4 described in gottlieb et al 2011 the backward euler and trapezoidal implicit methods are adopted in the full and semi implicit schemes as proposed in eyre unpublished manufactured solutions for one dimensional problems are used in order to evaluate the errors and to compare the different numerical methods by choosing an adequate discretization for ac equations resulting from the previous analysis of the first part of the work in the second part we propose a numerical semi implicit scheme to solve the damage and fracture model described in boldrini et al 2016 this procedure employs the fem for spatial discretization the newmark method for time integration of the kinematics equation and the backward euler for the damage phase field evolution finally results for 2d benchmark tests are presented for the fracture phase field model and the convergence to a sharp crack for a small width of the damage phase field layer is verified

an adaptive finite element approximation of a variational model of brittle fracture the energy of the francfort marigo model of brittle fracture can be approximated in the sense of gamma convergence by the ambrosio tortorelli functional in this work we formulate and analyze two adaptive finite element algorithms for the computation of its local minimizers for each algorithm we combine a newton type method with residual driven adaptive mesh refinement we present two theoretical results which demonstrate convergence of our algorithms to local minimizers of the ambrosio tortorelli functional

phase field modeling of ductile fracture at finite strains a robust variational based numerical implementation of a gradient extended theory by micromorphic regularization summary this work provides a robust variational based numerical implementation of a phase field model of ductile fracture in elastic plastic solids undergoing large strains this covers a computationally efficient micromorphic regularization of the coupled gradient plasticity damage formulation the phase field approach regularizes sharp crack surfaces within a pure continuum setting by a specific gradient damage modeling with geometric features rooted in fracture mechanics it has proven immensely successful with regard to the analysis of complex crack topologies without the need for fracture specific computational structures such as finite element design of crack discontinuities or intricate crack tracking algorithms the proposed gradient extended plasticity damage formulation includes two independent length scales that regularize both the plastic response as well as the crack discontinuities this ensures that the damage zones of ductile fracture are inside of plastic zones or vice versa and guarantees on the computational side a mesh objectivity in post critical ranges the proposed setting is rooted in a canonical variational principle the coupling of gradient plasticity to gradient damage is realized by a constitutive work density function that includes the stored elastic energy and the dissipated work due to plasticity and fracture the latter represents a coupled resistance to plasticity and damage depending on the gradient extended internal variables that enter plastic yield functions and fracture threshold functions with this viewpoint on the generalized internal variables at hand the thermodynamic formulation is outlined for gradient extended dissipative solids with generalized internal variables that are passive in nature it is specified for a conceptual model of von mises type elasto plasticity at finite strains coupled with fracture the canonical theory proposed is shown to be governed by a rate type minimization principle which fully determines the coupled multi field evolution problem this is exploited on the numerical side by a fully symmetric monolithic finite element implementation an important aspect of this work is the regularization towards a micromorphic gradient plasticity damage setting by taking into account additional internal variable fields linked to the original ones by penalty terms this enhances the robustness of the finite element implementation in particular on the side of gradient plasticity the performance of the formulation is demonstrated by means of some representative examples copyright 2016 john wiley sons ltd

quasifinite representations of classical lie subalgebras of w p we show that there are exactly two anti involutions of the algebra of differential operators on the circle that are a multiple of p t t preserving the principal gradation p c x non constant we classify the irreducible quasifinite highest weight representations of the central extension dp of the lie subalgebra fixed by the most important cases are the subalgebras dx of w that are obtained when p x x in these cases we realize the irreducible quasifinite highest weight modules in terms of highest weight representation of the central extension of the lie algebra of infinite matrices with finitely many nonzero diagonals over the algebra c u um 1 and its classical lie subalgebras of c and d types

a higher order phase field model for brittle fracture formulation and analysis within the isogeometric analysis framework phase field models based on the variational formulation for brittle fracture have recently been gaining popularity these models have proven capable of accurately and robustly predicting complex crack behavior in both two and three dimensions in this work we propose a fourth order model for the phase field approximation of the variational formulation for brittle fracture we derive the thermodynamically consistent governing equations for the fourth order phase field model by way of a variational principle based on energy balance assumptions the resulting model leads to higher regularity in the exact phase field solution which can be exploited by the smooth spline function spaces utilized in isogeometric analysis this increased regularity improves the convergence rate of the numerical solution and opens the door to higher order convergence rates for fracture problems we present an analysis of our proposed theory and numerical examples that support this claim we also demonstrate the robustness of the model in capturing complex three dimensional crack behavior

finite growth representations of infinite lie conformal algebras we classify all finite growth representations of all infinite rank subalgebras of the lie conformal algebra gc1 that contain a virasoro subalgebra

qhwm of the orthogonal type lie subalgebra of the lie algebra of matrix differential operators on the circle in this paper we classify the irreducible quasifinite highest weight modules of the orthogonal lie subalgebra of the lie algebra of matrix differential operators on the circle we also realize them in terms of the representation theory of the complex lie algebra gl m of infinite matrices with a finite number of nonzero diagonals with entries in the algebra of truncated polynomials and the corresponding subalgebras of type b and d

polynomial relations in the heisenberg algebra polynomial relations between the generators of the classical and quantum heisenberg algebras are presented some of those relations can have the meaning of the formulas of the normal ordering for the creation annihilation operators which occur in the method of the second quantization polynomial relations of a special type invariant under quantization and q deformation are discovered

liquid vapor transformations with surfactants phase field model and isogeometric analysis surfactants are compounds that find energetically favorable to be located at the boundaries between fluids they are able to modify the properties of those interfaces for example reducing surface tension here we propose a new model for liquid vapor flows with surfactants which captures the dynamics of the surfactant and accounts for phase transformations in the fluid the aforementioned model is derived from a free energy functional by using a coleman noll approach the proposed theory emanates from the isothermal navier stokes korteweg equations which describe single component two phase flow and naturally allow for phase transformations we believe that our model has significant potential to study the influence of surfactants in vaporization and condensation processes from a numerical point of view the proposed model poses significant challenges to existing discretization methods including stiffness in space and time internal and boundary layers as well as higher order partial differential operators to overcome these challenges we propose algorithms based on isogeometric analysis which permit an accurate and efficient discretization finally we illustrate the viability of the theoretical framework and the effectiveness of our algorithms by solving several numerical problems in two and three dimensions

the logarithm of the derivative operator and higher spin algebras of w type we use the notion of the logarithm of the derivative operator to describew type algebras as central extensions of the algebra of differential operators we also provide closed formulae for the truncations ofw1 to higher spin algebras withs m for allm 2 the results are extended to matrix valued differential operators introducing a logarithmic generalization of the maurer cartan cocycle

isogeometric finite element data structures based on bezier extraction of t splines we develop finite element data structures for t splines based on bezier extraction generalizing our previous work for nurbs as in traditional finite element analysis the extracted bezier elements are defined in terms of a fixed set of polynomial basis functions the so called bernstein basis the bezier elements may be processed in the same way as in a standard finite element computer program utilizing exactly the same data processing arrays in fact only the shape function subroutine needs to be modified while all other aspects of a finite element program remain the same a byproduct of the extraction process is the element extraction operator this operator localizes the topological and global smoothness information to the element level and represents a canonical treatment of t junctions referred to as hanging nodes in finite element analysis and a fundamental feature of t splines a detailed example is presented to illustrate the ideas copyright 2011 john wiley sons ltd

on the crack driving force of phase field models in linearized and finite elasticity abstract the phase field approach to fracture has been proven to be a mathematically sound and easy to implement method for computing crack propagation with arbitrary crack paths hereby crack growth is driven by energy minimization resulting in a variational crack driving force the definition of this force out of a tension related energy functional does however not always agree with the established failure criteria of fracture mechanics in this work different variational formulations for linear and finite elastic materials are discussed and ad hoc driving forces are presented which are motivated by general fracture mechanical considerations the superiority of the generalized approach is demonstrated by a series of numerical examples

γ convergence for high order phase field fracture continuum and isogeometric formulations abstract we consider high order phase field functionals introduced in borden et al 2014 and provide a rigorous proof that these functionals converge to a sharp crack brittle fracture energy we take into account three dimensional problems in linear elastic fracture mechanics and functionals defined both in sobolev spaces and in spaces of tensor product b splines in the latter convergence holds when the mesh size vanishes faster than the internal length of the phase field model on the theoretical level this condition is natural since the size of the phase field layer around the crack itself scales like the internal length on the numerical level it should be satisfied by local h refinement technically convergence holds in the sense of convergence with respect to the strong topology of l 1 while the sharp crack energy is defined in g s b d 2 the constraint on the phase field to take values in 0 1 is taken into account both in the sobolev setting and in the iso geometric setting in the latter it requires a special treatment since the projection operator on the space of tensor product b splines is not lagrangian i e interpolatory

on the complexity of integer programming a simple proof that integer programming ts in x s given the proof also estabhshes that there s a pseudopolynomial tune algorithm for integer programmmg with any fixed number of constraints

on approximation algorithms for concave mixed integer quadratic programming concave mixed integer quadratic programming is the problem of minimizing a concave quadratic polynomial over the mixed integer points in a polyhedral region in this work we describe an algorithm that finds an epsilon approximate solution to a concave mixed integer quadratic programming problem the running time of the proposed algorithm is polynomial in the size of the problem and in 1 epsilon provided that the number of integer variables and the number of negative eigenvalues of the objective function are fixed the running time of the proposed algorithm is expected unless mathcal p mathcal np

a higher order nonlocal operator method for solving partial differential equations abstract a higher order nonlocal operator method for the solution of boundary value problems is developed the proposed higher order nonlocal operator brings several advantages as compared to the original nonlocal operator method ren et al 2020 which only ensures first order convergence furthermore it can be applied to directly and efficiently obtain all partial derivatives of higher orders simultaneously without the need of using shape functions only the functionals based on the nonlocal operators termed as operator functional are needed to obtain the final discrete system of equations which significantly facilitates the implementation several numerical examples are presented to show the effectiveness and accuracy of the proposed higher order nonlocal operator method including the solution of the poisson equation in 2 5 dimensional space kirchhoff and von karman plate problems incompressible elastic materials as well as phase field modeling of fracture

on penalization in variational phase field models of brittle fracture abstract irreversible evolution is one of the central concepts as well as implementation challenges of both the variational approach to fracture by francfort and marigo 1998 and its regularized counterpart by bourdin francfort and marigo 2000 2007 and 2008 which is commonly referred to as a phase field model of brittle fracture irreversibility of the crack phase field imposed to prevent fracture healing leads to a constrained minimization problem whose optimality condition is given by a variational inequality in our study the irreversibility is handled via penalization provided the penalty constant is well tuned the penalized formulation is a good approximation to the original one with the advantage that the induced equality based weak problem enables a much simpler algorithmic treatment we propose an analytical procedure for deriving the optimal penalty constant more precisely its lower bound which guarantees a sufficiently accurate enforcement of the crack phase field irreversibility our main tool is the notion of the optimal phase field profile as well as the convergence result it is shown that the explicit lower bound is a function of two formulation parameters the fracture toughness and the regularization length scale but is independent of the problem setup geometry boundary conditions etc the formulation ingredients degradation function tension compression split etc as well as the discretization mesh size the optimally penalized formulation is tested for three benchmark problems including one with available analytical solution we also compare our results with those obtained by using the alternative irreversibility technique based on the notion of the history field by miehe et al 2010

some np complete problems in quadratic and nonlinear programming in continuous variable smooth nonconvex nonlinear programming we analyze the complexity of checking whether a a given feasible solution is not a local minimum and b the objective function is not bounded below on the set of feasible solutions we construct a special class of indefinite quadratic programs with simple constraints and integer data and show that checking a or b on this class is np complete as a corollary we show that checking whether a given integer square matrix is not copositive is np complete

a phase field description of dynamic brittle fracture abstract in contrast to discrete descriptions of fracture phase field descriptions do not require numerical tracking of discontinuities in the displacement field this greatly reduces implementation complexity in this work we extend a phase field model for quasi static brittle fracture to the dynamic case we introduce a phase field approximation to the lagrangian for discrete fracture problems and derive the coupled system of equations that govern the motion of the body and evolution of the phase field we study the behavior of the model in one dimension and show how it influences material properties for the temporal discretization of the equations of motion we present both a monolithic and staggered time integration scheme we study the behavior of the dynamic model by performing a number of two and three dimensional numerical experiments we also introduce a local adaptive refinement strategy and study its performance in the context of locally refined t splines we show that the combination of the phase field model and local adaptive refinement provides an effective method for simulating fracture in three dimensions

optimal approximations by piecewise smooth functions and associated variational problems abstract this reprint will introduce and study the most basic properties of three new variational problems which are suggested by applications to computer vision in computer vision a fundamental problem is to appropriately decompose the domain r of a function g x y of two variables this problem starts by describing the physical situation which produces images assume that a three dimensional world is observed by an eye or camera from some point p and that g1 rho represents the intensity of the light in this world approaching the point sub 1 from a direction rho if one has a lens at p focusing this light on a retina or a film in both cases a plane domain r in which we may introduce coordinates x y then let g x y be the strength of the light signal striking r at a point with coordinates x y g x y is essentially the same as sub 1 rho possibly after a simple transformation given by the geometry of the imaging syste the function g x y defined on the plane domain r will be called an image what sort of function is g the light reflected off the surfaces si of various solid objects o sub i visible from p will strike the domain r in various open subsets r sub i when one object o1 is partially in front of another object o2 as seen from p but some of object o2 appears as the background to the sides of o1 then the open sets r1 and r2 will have a common boundary the edge of object o1 in the image defined on r and one usually expects the image g x y to be discontinuous along this boundary jhd

a phase field model for rate independent crack propagation robust algorithmic implementation based on operator splits the computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies this can be overcome by a diffusive crack modeling based on the introduction of a crack phase field following our recent work c miehe f welschinger m hofacker thermodynamically consistent phase field models of fracture variational principles and multi field fe implementations international journal for numerical methods in engineering doi 10 1002 nme 2861 on phase field type fracture we propose in this paper a new variational framework for rate independent diffusive fracture that bases on the introduction of a local history field it contains a maximum reference energy obtained in the deformation history which may be considered as a measure for the maximum tensile strain obtained in history it is shown that this local variable drives the evolution of the crack phase field the introduction of the history field provides a very transparent representation of the balance equation that governs the diffusive crack topology in particular it allows for the construction of a new algorithmic treatment of diffusive fracture here we propose an extremely robust operator split scheme that successively updates in a typical time step the history field the crack phase field and finally the displacement field a regularization based on a viscous crack resistance that even enhances the robustness of the algorithm may easily be added the proposed algorithm is considered to be the canonically simple scheme for the treatment of diffusive fracture in elastic solids we demonstrate the performance of the phase field formulation of fracture by means of representative numerical examples

the tetrahedral finite cell method higher order immersogeometric analysis on adaptive non boundary fitted meshes the finite cell method fcm is an immersed domain finite element method that combines higher order non boundary fitted meshes weak enforcement of dirichlet boundary conditions and adaptive quadrature based on recursive subdivision due to its ability to improve the geometric resolution of intersected elements it can be characterized as an immersogeometric method in this paper we extend the fcm so far only used with cartesian hexahedral elements to higher order non boundary fitted tetrahedral meshes based on a reformulation of the octree based subdivision algorithm for tetrahedral elements we show that the resulting tetfcm scheme is fully accurate in an immersogeometric sense that is the solution fields achieve optimal and exponential rates of convergence for h and p refinement if the immersed geometry is resolved with sufficient accuracy tetfcm can leverage the natural ability of tetrahedral elements for local mesh refinement in three dimensions its suitability for problems with sharp gradients and highly localized features is illustrated by the immersogeometric phase field fracture analysis of a human femur bone

as a discretization of virasoro algebra it is shown that a realization of algebra in the kp hierarchy is equivalent to a difference operator realization of the simplest subalgebra of a q discretized virasoro algebra d to this end we first introduce the non commutative flow of the kp hierarchy and see the additional flow corresponds to the element of the algebra it is then shown that the new flow constructed by use of the additional one can be recognized as an action of d in the fock representation of the kp hierarchy

an interactive geometry modeling and parametric design platform for isogeometric analysis in this paper an interactive parametric design through analysis platform is proposed to help design engineers and analysts make more effective use of isogeometric analysis iga to improve their product design and performance we develop several rhinoceros rhino plug ins to take input design parameters through a user friendly interface generate appropriate surface and or volumetric models perform mechanical analysis and visualize the solution fields all within the same computer aided design cad program as part of this effort we propose and implement graphical generative algorithms for iga model creation and visualization based on grasshopper a visual programming interface to rhino the developed platform is demonstrated on two structural mechanics examples an actual wind turbine blade and a model of an integrally bladed rotor ibr in the latter example we demonstrate how the rhino functionality may be utilized to create conforming volumetric models for iga

character and determinant formulae of quasifinite representation of thew1 algebra we diagonalize the hilbert space of some subclass of the quasifinite module of thew1 algebra states are classified according to their eigenvalues for infinitely many commuting charges and the young diagrams the parameter dependence of their norms is explicitly derived the full character formulae of the degenerate representations are given as summation of the bilinear combinations of the schur polynomials

a fractional phase field model for two phase flows with tunable sharpness algorithms and simulations abstract we develop a fractional extension of a mass conserving allen cahn phase field model that describes the mixture of two incompressible fluids the fractional order controls the sharpness of the interface which is typically diffusive in integer order phase field models the model is derived based on an energy variational formulation an additional constraint is employed to make the allen cahn formulation mass conserving and comparable to the cahn hilliard formulation but at reduced cost the spatial discretization is based on a petrov galerkin spectral method whereas the temporal discretization is based on a stabilized adi scheme both for the phase field equation and for the navier stokes equation we demonstrate the spectral accuracy of the method with fabricated smooth solutions and also the ability to control the interface thickness between two fluids with different viscosity and density in simulations of two phase flow in a pipe and of a rising bubble we also demonstrate that using a formulation with variable fractional order we can deal simultaneously with both erroneous boundary effects and sharpening of the interface at no extra resolution

isogeometric collocation for phase field fracture models abstract phase field models based on the variational formulation for brittle fracture have recently been shown capable of accurately and robustly predicting complex crack behavior their numerical implementation requires costly operations at the quadrature point level which may include finding eigenvalues and forming tensor projection operators we explore the application of isogeometric collocation methods for the discretization of second order and fourth order phase field fracture models we show that a switch from isogeometric galerkin to isogeometric collocation methods has the potential to significantly speed up phase field fracture computations due to a reduction of point evaluations we advocate a hybrid collocation galerkin formulation that provides a consistent way of weakly enforcing neumann boundary conditions and multi patch interface constraints is able to handle the multiple boundary integral terms that arise from the weighted residual formulation and offers the flexibility to adaptively improve the crack resolution in the fracture zone we present numerical examples in one and two dimensions that illustrate the advantages of our approach

chva tal closures for mixed integer programming problems chvatal introduced the idea of viewing cutting planes as a system for proving that every integral solution of a given set of linear inequalities satisfies another given linear inequality this viewpoint has proven to be very useful in many studies of combinatorial and integer programming problems the basic ingredient in these cutting plane proofs is that for a polyhedronp and integral vectorw if max wx x p wx integer t thenwx t is valid for all integral vectors inp we consider the variant of this step where the requirement thatwx be integer may be replaced by the requirement that bar wx be integer for some other integral vector bar w the cutting plane proofs thus obtained may be seen either as an abstraction of gomory s mixed integer cutting plane technique or as a proof version of a simple class of the disjunctive cutting planes studied by balas and jeroslow our main result is that for a given polyhedronp the set of vectors that satisfy every cutting plane forp with respect to a specified subset of integer variables is again a polyhedron this allows us to obtain a finite recursive procedure for generating the mixed integer hull of a polyhedron analogous to the process of repeatedly taking chvatal closures in the integer programming case these results are illustrated with a number of examples from combinatorial optimization our work can be seen as a continuation of that of nemhauser and wolsey on mixed integer cutting planes

modified newton methods for solving fully monolithic phase field quasi static brittle fracture propagation abstract our goal in this work is to develop and compare modified newton methods for fully monolithic quasi static brittle phase field fracture propagation in variational phase field fracture a smoothed phase field indicator variable denoting the crack path is coupled to elasticity moreover a crack irreversibility condition is incorporated to develop a fully monolithic scheme is an extremely challenging task since the underlying problem is non convex and the jacobian of newton s method is indefinite to split the problem using alternating minimization thus a partitioned approach is a possible resort however there are good reasons to consider monolithic approaches such as for example robustness and efficiency although an error oriented newton method can cope with a larger variety of configurations it appears that this method is not always robust and also not always efficient inspired by nonlinear flow problems as alternative we develop a modified newton scheme in which globalization is based on a dynamic modification of the jacobian matrix rather than utilizing line search or trust region strategies this variation switches smoothly between full newton and newton like steps in several 2d and 3d numerical examples all of them with different characteristic features our modified newton solver is compared to a backtracking line search newton method another line search method monitoring the global energy and allowing for negative curvatures and to already published results of an error oriented version these computations also include further modifications of newton s method and detailed discussions why certain schemes either work or fail revisiting all findings the main outcome of this paper is that the modified newton scheme with jacobian modification is currently the only method that works in a robust and efficient way for all provided examples whereas line search schemes or the error oriented scheme show deficiencies for certain configurations

a phase field model for cohesive fracture in this paper a phase field model for cohesive fracture is developed after casting the cohesive zone approach in an energetic framework which is suitable for incorporation in phase field approaches the phase field approach to brittle fracture is recapitulated the approximation to the dirac function is discussed with particular emphasis on the dirichlet boundary conditions that arise in the phase field approximation the accuracy of the discretisation of the phase field including the sensitivity to the parameter that balances the field and the boundary contributions is assessed at the hand of a simple example the relation to gradient enhanced damage models is highlighted and some comments on the similarities and the differences between phase field approaches to fracture and gradient damage models are made a phase field representation for cohesive fracture is elaborated starting from the aforementioned energetic framework the strong as well as the weak formats are presented the latter being the starting point for the ensuing finite element discretisation which involves three fields the displacement field an auxiliary field that represents the jump in the displacement across the crack and the phase field compared to phase field approaches for brittle fracture the modelling of the jump of the displacement across the crack is a complication and the current work provides evidence that an additional constraint has to be provided in the sense that the auxiliary field must be constant in the direction orthogonal to the crack the sensitivity of the results with respect to the numerical parameter needed to enforce this constraint is investigated as well as how the results depend on the orders of the discretisation of the three fields finally examples are given that demonstrate grid insensitivity for adhesive and for cohesive failure the latter example being somewhat limited because only straight crack propagation is considered

subalgebras of w 1 infinity and their quasifinite representations we propose a series of new subalgebras of the w1 infinity algebra parametrized by polynomials p omega and study their quasifinite representations we also investigate the relation between such subalgebras and the gl infinity algebra as an example we investigate the winfinity algebra which corresponds to the case p omega w presenting its free field realizations and kac determinants at lower levels

ellipsoidal mixed integer representability representability results for mixed integer linear systems play a fundamental role in optimization since they give geometric characterizations of the feasible sets that can be formulated by mixed integer linear programming we consider a natural extension of mixed integer linear systems obtained by adding just one ellipsoidal inequality the set of points that can be described possibly using additional variables by these systems are called ellipsoidal mixed integer representable in this work we give geometric conditions that characterize ellipsoidal mixed integer representable sets

sensitivity and uncertainty analysis for flexoelectric nanostructures abstract in this paper sensitivity analysis has been applied to identify the key input parameters influencing the energy conversion factor ecf of flexoelectric materials the governing equations of flexoelectricity are modeled by a nurbs based iga formulation exploiting their higher order continuity and hence avoiding a complex mixed formulation the examined input parameters include model and material properties and the sampling has been obtained using the latin hypercube sampling lhs method in the probability space the sensitivity of the model output to each of the input parameters at different aspect ratios of the beam is quantified by three various common methods i e morris one at a time moat pce sobol and extended fourier amplitude sensitivity test efast the numerical results indicate that the flexoelectric constants are the most dominant factors influencing the uncertainties in the energy conversion factor in particular the transversal flexoelectric coefficient h 12 moreover the model parameters also show considerable interaction effects of the material properties

a higher order stress based gradient enhanced damage model based on isogeometric analysis abstract the micro damage associated with diffuse fracture processes in quasi brittle materials can be described by continuum damage mechanics in order to overcome the mesh dependence of local damage formulations non local and gradient enhanced approaches are often employed in this manuscript a higher order stress based gradient enhanced formulation is proposed which exploits the higher order continuity of b spline functions in isogeometric analysis iga the proposed formulation does not require the decomposition of the fourth order model into two second order models two numerical examples are presented to demonstrate the performance of the formulation and to compare the obtained solutions with results from conventional gradient enhanced damage formulation

fptas for optimizing polynomials over the mixed integer points of polytopes in fixed dimension we show the existence of a fully polynomial time approximation scheme fptas for the problem of maximizing a non negative polynomial over mixed integer sets in convex polytopes when the number of variables is fixed moreover using a weaker notion of approximation we show the existence of a fully polynomial time approximation scheme for the problem of maximizing or minimizing an arbitrary polynomial over mixed integer sets in convex polytopes when the number of variables is fixed

superconvergent isogeometric free vibration analysis of euler bernoulli beams and kirchhoff plates with new higher order mass matrices abstract a superconvergent isogeometric free vibration analysis is presented for euler bernoulli beams and kirchhoff plates this method is featured by new higher order mass matrices for the 1d euler bernoulli beam problem it is shown that a new higher order mass matrix can be directly established by optimizing a reduced bandwidth mass matrix the reduced bandwidth mass matrix is designed based upon the consistent mass matrix and it contains adjustable parameters to be determined via maximizing the order of accuracy of the vibration frequency as a result 4th and 6th orders of accuracy are observed for the proposed quadratic and cubic higher order mass matrices while their corresponding consistent mass matrices are 2nd and 4th order accurate respectively thus the higher order beam mass matrices have more superior frequency accuracy simultaneously with smaller bandwidth compared with their corresponding consistent mass matrices while for the 2d kirchhoff plate problem in order to compute arbitrary frequency in a superconvergent fashion a mixed mass matrix is formulated through a linear combination of the consistent mass matrix and the reduced bandwidth mass matrix then by introducing the wave propagation angle the higher order plate mass matrix can be rationally derived from the mixed mass matrix it is proved that the optimal combination parameter for higher order mass matrix depends on the wave propagation angle consequently a particular higher order mass matrix can always be set up for a superconvergent computation of arbitrary vibration frequency it is shown that the quadratic and cubic higher order mass matrices possess 4th and 6th orders of accuracy which are two orders higher than those of the consistent plate mass matrices the construction of higher order mass matrices and the analytical results for vibration frequencies are systematically demonstrated by a set of numerical examples

minimizing cubic and homogeneous polynomials over integers in the plane we complete the complexity classification by degree of minimizing a polynomial over the integer points in a polyhedron in a real vector space of dimension two previous work shows that optimizing a quadratic polynomial over the integer points in a polyhedral region in a real vector space of dimension two can be done in polynomial time whereas optimizing a quartic polynomial in the same type of region is np hard we close the gap by showing that this problem can be solved in polynomial time for cubic polynomials furthermore we show that the problem of minimizing a homogeneous polynomial of any fixed degree over the integer points in a bounded polyhedron in a real vector space of dimension two is solvable in polynomial time we show that this holds for polynomials that can be translated into homogeneous polynomials even when the translation vector is unknown we demonstrate that such problems in the unbounded case can have smallest optimal solutions of exponential size in the size of the input thus req

quasifinite highest weight modules over the lie algebra of differential operators on the circle we classify positive energy representations with finite degeneracies of the lie algebraw 1 and construct them in terms of representation theory of the lie algebra hat gl infty r m of infinites matrices with finite number of non zero diagonals over the algebrar m t t m 1 the unitary ones are classified as well similar results are obtained for the sin algebras

fourth order phase field model for local max ent approximants applied to crack propagation we apply a fourth order phase field model for fracture based on local maximum entropy lme approximants the higher order continuity of the meshfree lme approximants allows to directly solve the fourth order phase field equations without splitting the fourth order differential equation into two second order differential equations we will first show that the crack surface can be captured more accurately in the fourth order model furthermore less nodes are needed for the fourth order model to resolve the crack path finally we demonstrate the performance of the proposed meshfree fourth order phase field formulation for 5 representative numerical examples computational results will be compared to analytical solutions within linear elastic fracture mechanics and experimental data for three dimensional crack propagation

efficient numerical methods for simulating surface tension of multi component mixtures with the gradient theory of fluid interfaces abstract surface tension significantly impacts subsurface flow and transport and it is the main cause of capillary effect a major immiscible two phase flow mechanism for systems with a strong wettability preference in this paper we consider the numerical simulation of the surface tension of multi component mixtures with the gradient theory of fluid interfaces major numerical challenges include that the system of the euler lagrange equations is solved on the infinite interval and the coefficient matrix is not positive definite we construct a linear transformation to reduce the euler lagrange equations and naturally introduce a path function which is proven to be a monotonic function of the spatial coordinate variable by using the linear transformation and the path function we overcome the above difficulties and develop the efficient methods for calculating the interface and its interior compositions moreover the computation of the surface tension is also simplified the proposed methods do not need to solve the differential equation system and they are easy to be implemented in practical applications numerical examples are tested to verify the efficiency of the proposed methods