a gradient reproducing kernel collocation method for boundary value problems summary the earlier work in the development of direct strong form collocation methods such as the reproducing kernel collocation method rkcm addressed the domain integration issue in the galerkin type meshfree method such as the reproducing kernel particle method but with increased computational complexity because of taking higher order derivatives of the approximation functions and the need for using a large number of collocation points for optimal convergence in this work we intend to address the computational complexity in rkcm while achieving optimal convergence by introducing a gradient reproduction kernel approximation the proposed gradient rkcm reduces the order of differentiation to the first order for solving second order pdes with strong form collocation we also show that different from the typical strong form collocation method where a significantly large number of collocation points than the number of source points is needed for optimal convergence the same number of collocation points and source points can be used in gradient rkcm we also show that the same order of convergence rates in the primary unknown and its first order derivative is achieved owing to the imposition of gradient reproducing conditions the numerical examples are given to verify the analytical prediction copyright 2012 john wiley sons ltd

lifting and separation procedures for the cut polytope the max cut problem and the associated cut polytope on complete graphs have been extensively studied over the last 25 years however in comparison only little research has been conducted for the cut polytope on arbitrary graphs in particular separation algorithms have received only little attention in this study we describe new separation and lifting procedures for the cut polytope on general graphs these procedures exploit algorithmic and structural results known for the cut polytope on complete graphs to generate valid and sometimes facet defining inequalities for the cut polytope on arbitrary graphs in a cutting plane framework we report computational results on a set of well established benchmark problems

locking free isogeometric collocation methods for spatial timoshenko rods in this work we present the application of isogeometric collocation techniques to the solution of spatial timoshenko rods the strong form equations of the problem are presented in both displacement based and mixed formulations and are discretized via nurbs based isogeometric collocation several numerical experiments are reported to test the accuracy and efficiency of the considered methods as well as their applicability to problems of practical interest in particular it is shown that mixed collocation schemes are locking free independently of the choice of the polynomial degrees for the unknown fields such an important property is also analytically proven

an isogeometric collocation approach for bernoulli euler beams and kirchhoff plates abstract in this paper iga collocation methods are for the first time introduced for the solution of thin structural problems described by the bernoulli euler beam and kirchhoff plate models in particular a precise description of the proposed methods of the relevant implementation details and of the strategy to efficiently deal with different combinations of boundary conditions is given finally several numerical experiments confirm that the proposed formulations represent an efficient and geometrically flexible tool for the simulation of thin structures

on the cut polytope the cut polytopep c g of a graphg v e is the convex hull of the incidence vectors of all edge sets of cuts ofg we show some classes of facet defining inequalities ofp c g we describe three methods with which new facet defining inequalities ofp c g can be constructed from known ones in particular we show that inequalities associated with chordless cycles define facets of this polytope moreover for these inequalities a polynomial algorithm to solve the separation problem is presented we characterize the facet defining inequalities ofp c g ifg is not contractible tok 5 we give a simple characterization of adjacency inp c g and prove that for complete graphs this polytope has diameter one and thatp c g has the hirsch property a relationship betweenp c g and the convex hull of incidence vectors of balancing edge sets of a signed graph is studied

isogeometric collocation neumann boundary conditions and contact abstract isogeometric collocation methods have been proposed recently and their accuracy and efficiency demonstrated for elastostatics and explicit dynamics this paper addresses two important aspects in the development of the isogeometric collocation technology namely the imposition of neumann boundary conditions and the enforcement of contact constraints which are both treated within the same framework it is shown that the strong imposition of neumann boundary conditions may lead to a significant loss of accuracy in some situations in particular when non uniform meshes are used two possible remedies are proposed to restore the desired level of accuracy while keeping the computational cost virtually unchanged i e a hybrid collocation galerkin approach and an enhanced collocation ec approach a frictionless contact formulation suitable for the collocation framework is further proposed and shown to pass the contact patch test to machine precision when combined with the ec approach the formulation is shown to deliver accurate results and to perform robustly also for highly non uniform meshes for all the collocation formulations contact pressures are greater than or equal to zero pointwise in contrast with standard lagrange finite elements

explicit isogeometric collocation for the dynamics of three dimensional beams undergoing finite motions abstract we initiate the study of three dimensional shear deformable geometrically exact beam dynamics through explicit isogeometric collocation methods the formulation we propose is based on a natural combination of the chosen finite rotations representation with an explicit geometrically consistent lie group time integrator we focus on extending the integration scheme originally proposed for rigid body dynamics to our nonlinear initial boundary value problem where special attention is required by neumann boundary conditions the overall formulation is simple and only relies on a geometrically consistent procedure to compute the internal forces once control angular and linear accelerations of the beam cross sections are obtained from the previous time step the capabilities of the method are shown through numerical applications involving very large displacements and rotations and different boundary conditions

a simple algorithm for obtaining nearly optimal quadrature rules for nurbs based isogeometric analysis we develop new quadrature rules for isogeometric analysis based on the solution of a local nonlinear problem a simple and robust algorithm is developed to determine the rules which are exact for important b spline spaces of uniform and geometrically stretched knot spacings we consider both periodic and open knot vector configurations and illustrate the efficiency of the rules on selected boundary value problems we find that the rules are almost optimally efficient but much easier to obtain than optimal rules which require the solution of global nonlinear problems that are often ill posed

isogeometric collocation for elastostatics and explicit dynamics we extend the development of collocation methods within the framework of isogeometric analysis iga to multi patch nurbs configurations various boundary and patch interface conditions and explicit dynamic analysis the methods developed are higher order accurate stable with no hourglass modes and efficient in that they require a minimum number of quadrature evaluations the combination of these attributes has not been obtained previously within standard finite element analysis

isogeometric collocation on planar multi patch domains abstract we present an isogeometric framework based on collocation to construct a c 2 smooth approximation of the solution of the poisson s equation over planar bilinearly parameterized multi patch domains the construction of the used globally c 2 smooth discretization space for the partial differential equation is simple and works uniformly for all possible multi patch configurations the basis of the c 2 smooth space can be described as the span of three different types of locally supported functions corresponding to the single patches edges and vertices of the multi patch domain for the selection of the collocation points which is important for the stability and convergence of the collocation problem two different choices are numerically investigated the first approach employs the tensor product greville abscissae as collocation points and shows for the multi patch case the same convergence behavior as for the one patch case 1 which is suboptimal in particular for odd spline degree the second approach generalizes the concept of superconvergent points from the one patch case cf 2 3 4 to the multi patch case again these points possess better convergence properties than greville abscissae in case of odd spline degree

a weighted point based formulation for isogeometric contact abstract in the context of isogeometric contact analysis surfaces of objects can be described smoothly due to the high continuity of the involved shape functions this facilitates construction of a continuous normal field which may be beneficial for modeling of contact problems this property allows for stable collocation of the virtual contact work using considerable less evaluation points than numerical integration while the approximation quality of the computational results is almost the same in this context a weighted point based method pts is introduced as an extension of the point to segment method pts proposed previously by matzen et al 2013 the weights enable transmission of contact stresses instead of contact forces along the contact surfaces in normal direction the non penetration condition is enforced by the lagrange multiplier method combined with the penalty method in tangential direction corresponding to the collocation weights a constant or nurbs lagrange multiplier field is introduced a comparison between the collocation based contact formulations pts pts and a mortar method is drawn showing that weighted collocation is able to achieve almost the same accuracy as the integration based mortar formulation

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

a point collocation method based on reproducing kernel approximations a reproducing kernel particle method with built in multiresolution features in a very attractive meshfree method for numerical solution of partial differential equations the design and implementation of a galerkin based reproducing kernel particle method however faces several challenges such as the issue of nodal volumes and accurate and efficient implementation of boundary conditions in this paper we present a point collocation method based on reproducing kernel approximations we show that in a point collocation approach the assignment of nodal volumes and implementation of boundary conditions are not critical issues and points can be sprinkled randomly making the point collocation method a true meshless approach the point collocation method based on reproducing kernel approximations however requires the calculation of higher order derivatives that would typically not be required in a galerkin method a correction function and reproducing conditions that enable consistency of the point collocation method are derived the point collocation method is shown to be accurate for several one and two dimensional problems and the convergence rate of the point collocation method is addressed copyright 2000 john wiley sons ltd

solving max cut to optimality by intersecting semidefinite and polyhedral relaxations we present a method for finding exact solutions of max cut the problem of finding a cut of maximum weight in a weighted graph we use a branch and bound setting that applies a dynamic version of the bundle method as bounding procedure this approach uses lagrangian duality to obtain a nearly optimal solution of the basic semidefinite max cut relaxation strengthened by triangle inequalities the expensive part of our bounding procedure is solving the basic semidefinite relaxation of the max cut problem which has to be done several times during the bounding process we review other solution approaches and compare the numerical results with our method we also extend our experiments to instances of unconstrained quadratic 0 1 optimization and to instances of the graph equipartition problem the experiments show that our method nearly always outperforms all other approaches in particular for dense graphs where linear programming based methods fail our method performs very well exact solutions are obtained in a reasonable time for any instance of size up to n 100 independent of the density for some problems of special structure we can solve even larger problem classes we could prove optimality for several problems of the literature where to the best of our knowledge no other method is able to do so

skew symmetric nitsche s formulation in isogeometric analysis dirichlet and symmetry conditions patch coupling and frictionless contact abstract a simple skew symmetric nitsche s formulation is introduced into the framework of isogeometric analysis iga to deal with various problems in small strain elasticity essential boundary conditions symmetry conditions for kirchhoff plates patch coupling in statics and in modal analysis as well as signorini contact conditions for linear boundary or interface conditions the skew symmetric formulation is parameter free for contact conditions it remains stable and accurate for a wide range of the stabilization parameter several numerical tests are performed to illustrate its accuracy stability and convergence performance we investigate particularly the effects introduced by nitsche s coupling including the convergence performance and condition numbers in statics as well as the extra outlier frequencies and corresponding eigenmodes in structural dynamics we present the hertz test the block test and a 3d self contact example showing that the skew symmetric nitsche s formulation is a suitable approach to simulate contact problems in iga

efficient semidefinite branch and cut for map mrf inference we propose a branch and cut b c method for solving general map mrf inference problems the core of our method is a very efficient bounding procedure which combines scalable semidefinite programming sdp and a cutting plane method for seeking violated constraints in order to further speed up the computation several strategies have been exploited including model reduction warm start and removal of inactive constraints we analyze the performance of the proposed method under different settings and demonstrate that our method either outperforms or performs on par with state of the art approaches especially when the connectivities are dense or when the relative magnitudes of the unary costs are low we achieve the best reported results experiments show that the proposed algorithm achieves better approximation than the state of the art methods within a variety of time budgets on challenging non submodular map mrf inference problems

uniruledness of strata of holomorphic differentials in small genus abstract we address the question concerning the birational geometry of the strata of holomorphic and quadratic differentials we show strata of holomorphic and quadratic differentials to be uniruled in small genus by constructing rational curves via pencils on k3 and del pezzo surfaces respectively restricting to genus 3 g 6 we construct projective bundles over a rational varieties that dominate the holomorphic strata with length at most g 1 hence showing in addition that these strata are unirational

hierarchical nurbs in frictionless contact abstract this work investigates mortar based frictionless contact in the context of nurbs discretizations that are subjected to local hierarchical refinement the investigations emphasize three sets of choices which lead to different contact algorithms that have distinct advantages and disadvantages first on the optimization side both exterior and interior point methods are applied thus spanning inexact constraint enforcement algorithms of the penalty or barrier type as well as exact ones of the primal dual type second on the discretization side the hierarchical basis set of the mortar variables is inherited either directly from the discretization of the slave surface or after an intermediate normalization step to satisfy the partition of unity third in interaction with both optimization and discretization the kinematic mortar variable is recovered from the actual normal gap through the full or lumped solution of a linear system of equations the implications of different choices are highlighted through benchmark problems which monitor the solution quality at the global level through the structural force evolution and at the local level through the contact pressure distribution

isogeometric large deformation frictionless contact using t splines t spline based isogeometric analysis is applied to frictionless contact problems between deformable bodies in the context of large deformations the continuum is discretized with cubic t splines and cubic nurbs a gauss point to surface formulation is combined with the penalty method to treat the contact constraints in the discretized setting it is demonstrated that analysis suitable t splines coupled with local refinement accurately approximate contact pressures with far fewer degrees of freedom than nurbs both two and three dimensional examples are presented additionally all t spline analysis models are generated using commercially available t spline modeling software without intermediate mesh generation or geometry clean up steps

a modified node to segment algorithm passing the contact patch test summary several investigations have shown that the classical one pass node to segment nts algorithms for the enforcement of contact constraints fail the contact patch test this implies that the algorithms may introduce solution errors at the contacting surfaces and these errors do not necessarily decrease with mesh refinement the previous research has mainly focused on the lagrange multiplier method to exactly enforce the contact geometry conditions the situation is even worse with the penalty method due to its inherent approximation that yields a solution affected by a non zero penetration the aim of this study is to analyze and improve the contact patch test behavior of the one pass nts algorithm used in conjunction with the penalty method for 2d frictionless contact the paper deals with the case of linear elements for this purpose several sequential modifications of the basic formulation have been considered which yield incremental improvements in results of the contact patch test the final proposed formulation is a modified one pass nts algorithm which is able to pass the contact patch test also if used in conjunction with the penalty method in other words this algorithm is able to correctly reproduce the transfer of a constant contact pressure with a constant proportional penetration copyright q 2009 john wiley sons ltd

a point to segment contact formulation for isogeometric nurbs based finite elements formulation of isogeometric finite elements has received a great deal of attention in the recent past the present study deals with the treatment of problems in structural mechanics including large elastic deformations and contact one decisive difference between isogeometric finite elements based on nurbs functions and standard finite elements fe based on lagrange polynomials is higher inter element continuity this is a promising characteristics to model contact problems as all issues associated with kinks between elements are naturally avoided particularly in cases with large sliding this has the potential to be an attractive feature we present a bilateral isogeometric collocation contact formulation for geometrically non linear two dimensional problems using greville and botella points to collocate the contact integrals different methods to obtain accurate and physically meaningful stress distributions are investigated and compared the results show that the higher inter element continuity in context of non smooth problems may imply undesired effects in the numerical solution such as unphysical stress oscillations similar effects have been observed in standard p fem numerical experiments indicate that these oscillations may be avoided if the basis functions of contact and non contact zones are separated by knot relocation and knot repetition

the geometry of the moduli space of odd spin curves the spin moduli spacesg is the parameter space of theta characteristics spin structures on stable curves of genus g it has two connected components s g ands g depending on the parity of the spin structure we establish a complete birational classication by kodaira dimension of the odd components g of the spin moduli space we show thats g is uniruled for g 12 and even unirational for g 8 in this range introducing the concept of cluster for the mukai variety whose one dimensional linear sections are general canonical curves of genus g we construct new birational models ofs g these we then use to explicitly describe the birational structure of s g for instance s 8 is birational to a locally trivial p 7 bundle over the moduli space of elliptic curves with seven pairs of marked points for g 12 we prove thats g is a variety of general type in genus 12 this requires the construction of a counterexample to the slope conjecture on eective divisors on the moduli space of stable curves of genus 12

shape optimization directly from cad an isogeometric boundary element approach using t splines abstract we develop a t spline isogeometric boundary element method igabem simpson et al 2012 scott et al 2013 simpson et al 2014 to shape sensitivity analysis and gradient based shape optimization in three dimensional linear elasticity contrary to finite element based isogeometric analysis iga approaches no parametrization of the volume is required hence the iterative optimization algorithm can be implemented directly from cad without any mesh generation or postprocessing step for returning the resulting structure to cad designers t splines also guarantee a water tight geometry without the manual geometrical repair work as with non uniform rational b splines nurbs we demonstrate the worth of the method by analysing problems with and without analytical solutions including engineering examples involving complex shapes additionally we provide all the derivations of the required sensitivities and the details pertaining to the geometries examined in the benchmarking to provide helpful reference problems for 3d shape optimization

an isogeometric collocation method using superconvergent points abstract we develop an iga collocation method modified by collocating at points other than the standard greville abscissae the method is related to orthogonal collocation used for solving differential equations and to the superconvergence theory therefore we refer to this method as super collocation iga sc by carefully choosing the collocation points it can be seen that the iga sc converges in the first derivative energy norms at rates similar to that of the galerkin solution this is different from the collocation at greville abscissae iga c where the convergence in energy norm for odd polynomial degrees is typically suboptimal the method is tested on 1d 2d and 3d numerical examples in which it is compared to iga c and galerkin s method iga g the comparison includes a detailed cost vs accuracy analysis which shows an improved efficiency of the proposed method in particular for odd polynomial degrees

a large deformation frictional contact formulation using nurbs based isogeometric analysis this paper focuses on the application of nurbs based isogeometric analysis to coulomb frictional contact problems between deformable bodies in the context of large deformations a mortar based approach is presented to treat the contact constraints whereby the discretization of the continuum is performed with arbitrary order nurbs as well as c0 continuous lagrange polynomial elements for comparison purposes the numerical examples show that the proposed contact formulation in conjunction with the nurbs discretization delivers accurate and robust predictions results of lower quality are obtained from the lagrange discretization as well as from a different contact formulation based on the enforcement of the contact constraints at every integration point on the contact surface copyright 2011 john wiley sons ltd

weighted radial basis collocation method for boundary value problems this work introduces the weighted radial basis collocation method for boundary value problems we first show that the employment of least squares functional with quadrature rules constitutes an approximation of the direct collocation method standard radial basis collocation method however yields a larger solution error near boundaries the residuals in the least squares functional associated with domain and boundary can be better balanced if the boundary collocation equations are properly weighted the error analysis shows unbalanced errors between domain neumann boundary and dirichlet boundary least squares terms a weighted least squares functional and the corresponding weighted radial basis collocation method are then proposed for correction of unbalanced errors it is shown that the proposed method with properly selected weights significantly enhances the numerical solution accuracy and convergence rates

point collocation methods using the fast moving least square reproducing kernel approximation a pseudo spectral point collocation meshfree method is proposed we apply a scheme of approximating derivatives based on the moving least square reproducing kernel approximations using approximated derivatives we propose a new point collocation method unlike other meshfree methods that require direct calculation of derivatives for shape functions with the proposed scheme approximated derivatives are obtained in the process of calculating the shape function itself without further cost moreover the scheme does not require the regularity of the window function which ensures the regularity of shape functions in this paper we show the reproducing property and the convergence of interpolation for approximated derivatives of shape functions as numerical examples of the proposed scheme poisson and stokes problems are considered in various situations including the case of randomly generated node sets in short the proposed scheme is efficient and accurate even for complicated geometry such as the flow past a cylinder copyright 2003 john wiley sons ltd

projective degenerations of k3 surfaces gaussian maps and fano threefolds in this article we exhibit certain projective degenerations of smoothk3 surfaces of degree 2g 2 in g whose picard group is generated by the hyperplane class to a union of two rational normal scrolls and also to a union of planes as a consequence we prove that the general hyperplane section of suchk3 surfaces has a corank one gaussian map ifg 11 org 13 we also prove that the general such hyperplane section lies on a uniquek3 surface up to projectivities finally we present a new approach to the classification of prime fano threefolds of index one which does not rely on the existence of a line

hybrid collocation galerkin approach for the analysis of surface represented 3d solids employing sb fem abstract this paper presents a numerical method to solve the three dimensional elasticity problem of surface represented solids a surface oriented formulation is derived in which the three dimensional solid is described by its boundary surfaces and a radial scaling center scaling the boundary surfaces with respect to the scaling center yields a parameterization of the complete solid the definition of the boundary surfaces is sufficient for the description of the solid thus the formulation conforms ideally to the boundary representation modeling technique used in cad in the present approach no tensor product structure of three dimensional objects is exploited to parameterize the physical domain the weak form of the equation of motion and the weak form of the neumann boundary conditions are enforced only at the boundary surfaces it leads to a transformation of the governing partial differential equations of elasticity to an ordinary differential equation ode of euler type solving the ode leads to the displacement in the radial scaling direction in the present approach the isogeometric galerkin finite element method is employed to describe the geometry of the boundary surfaces and also to approximate the displacement response of the boundary surfaces it exploits two dimensional nurbs objects to parameterize the boundary surfaces the final euler type ode is solved by the nurbs based collocation method the displacement response in the radial scaling direction is approximated by the nurbs basis functions hence the method can be extended to nonlinear problems the accuracy of the method is validated against analytical solutions in general the proposed formulation is able to model solids bounded by an arbitrary number of surfaces

an interior point method for semidefinite programming we propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices

a finite point method in computational mechanics applications to convective transport and fluid flow the paper presents a fully meshless procedure fo solving partial differential equations the approach termed generically the finite point method is based on a weighted least square interpolation of point data and point collocation for evaluating the approximation integrals some examples showing the accuracy of the method for solution of adjoint and non self adjoint equations typical of convective diffusive transport and also to the analysis of compressible fluid mechanics problem are presented

necessary conditions for extended noncontextuality in general sets of random variables in this paper we explore the graph approach to contextuality to restate the extended definition of noncontextuality as given by kujala et al phys rev lett 115 150401 2015 using graph theoretical terms this extended definition avoids the assumption of the pre sheaf or non disturbance condition which states that if two contexts overlap then the marginal distribution obtained for the intersection must be the same a restriction that will never be perfectly satisfied in real experiments with this we are able to derive necessary conditions for extended noncontextuality for any set of random variables based on the geometrical aspects of the graph approach which can be tested directly with experimental data in any contextuality experiment and which reduce to traditional necessary conditions for noncontextuality if the non disturbance condition is satisfied

generating facets for the cut polytope of a graph by triangular elimination the cut polytope of a graph arises in many fields although much is known about facets of the cut polytope of the complete graph very little is known for general graphs the study of bell inequalities in quantum information science requires knowledge of the facets of the cut polytope of the complete bipartite graph or more generally the complete k partite graph lifting is a central tool to prove certain inequalities are facet inducing for the cut polytope in this paper we introduce a lifting operation named triangular elimination applicable to the cut polytope of a wide range of graphs triangular elimination is a specific combination of zero lifting and fourier motzkin elimination using the triangle inequality we prove sufficient conditions for the triangular elimination of facet inducing inequalities to be facet inducing the proof is based on a variation of the lifting lemma adapted to general graphs the result can be used to derive facet inducing inequalities of the cut polytope of various graphs from those of the complete graph we also investigate the symmetry of facet inducing inequalities of the cut polytope of the complete bipartite graph derived by triangular elimination

least squares collocation meshless method a finite point method least squares collocation meshless method is proposed except for the collocation points which are used to construct the trial functions a number of auxiliary points are also adopted unlike the direct collocation method the equilibrium conditions are satisfied not only at the collocation points but also at the auxiliary points in a least squares sense the moving least squares interpolant is used to construct the trial functions the computational effort required for the present method is in the same order as that required for the direct collocation while the present method improves the accuracy of solution significantly the proposed method does not require any mesh so that it is a truly meshless method three numerical examples are studied in detail which show that the proposed method possesses high accuracy with low computational effort copyright 2001 john wiley sons ltd

moduli of nodal curves on k3 surfaces abstract we consider modular properties of nodal curves on general k3 surfaces let k p be the moduli space of primitively polarized k3 surfaces s l of genus p 3 and v p m k p be the universal severi variety of nodal irreducible curves in m l on s l k p we find conditions on p m for the existence of an irreducible component v of v p m on which the moduli map v m g with g m 2 p 1 1 has generically maximal rank differential our results which for any p leave only finitely many cases unsolved and are optimal for m 5 except for very low values of p are summarized in theorem 1 1 in the introduction

an application of combinatorial optimization to statistical physics and circuit layout design we study the problem of finding ground states of spin glasses with exterior magnetic field and the problem of minimizing the number of vias holes on a printed circuit board or contacts on a chip subject to pin preassignments and layer preferences the former problem comes up in solid state physics and the latter in very large scale integrated vlsi circuit design and in printed circuit board design both problems can be reduced to the max cut problem in graphs based on a partial characterization of the cut polytope we design a cutting plane algorithm and report on computational experience with it our method has been used to solve max cut problems on graphs with up to 1 600 nodes

three dimensional mortar based frictional contact treatment in isogeometric analysis with nurbs a three dimensional mortar based frictional contact treatment in isogeometric analysis with nurbs is presented in the finite deformation regime within a setting where the nurbs discretization of the contact surface is inherited directly from the nurbs discretization of the volume the contact integrals are evaluated through a mortar approach where the geometrical and frictional contact constraints are treated through a projection to control point quantities the formulation delivers a non negative pressure distribution and minimally oscillatory local contact interactions with respect to alternative lagrange discretizations independent of the discretization order these enable the achievement of improved smoothness in global contact forces and moments through higher order geometrical descriptions it is concluded that the presented mortar based approach serves as a common basis for treating isogeometric contact problems with varying orders of discretization throughout the contact surface and the volume

a collocated isogeometric finite element method based on gauss lobatto lagrange extraction of splines generalizing the concept of bezier extraction we introduce an extraction operator that links c0c0 gauss lobatto lagrange functions with smooth splines this opens the door for collocated isogeometric analysis that combines the accuracy of the galerkin method with collocation type formation and assembly procedures we present the key ingredients of the technology i e integration by parts and the weighted residual form the interaction of gauss lobatto lagrange extraction with gauss lobatto quadrature and symmetrization with the ultra weak formulation we compare the new method with standard isogeometric galerkin and isogeometric point collocation methods for spline discretizations in three dimensions

on cuts and matchings in planar graphs we study the max cut problem in graphs not contractible tok5 and optimum perfect matchings in planar graphs we prove that both problems can be formulated as polynomial size linear programs

effects of parameterization and knot placement techniques on primal and mixed isogeometric collocation formulations of spatial shear deformable beams with varying curvature and torsion abstract we present a displacement based and a mixed isogeometric collocation iga c formulation for free form three dimensional shear deformable beams with high and rapidly varying curvature and torsion when such complex shapes are concerned the approach used to build the iga geometric model becomes relevant although iga c has been so far successfully applied to a wide range of problems the effects that different parameterization and knot placement techniques may have on the accuracy of collocation based formulations is still an unexplored field to fill this gap primal and mixed formulations are used combining two parameterization methods chord length and equally spaced with two knot placement techniques uniformly spaced and de boor with respect to the space varying frenet local frame we derive the strong form of the governing equations in a compact form through the definition of two matrix operators conveniently used to perform first and second order derivatives of the vector fields involved in the formulations this approach is very efficient and easy to implement within a collocation based scheme several challenging numerical experiments allow to test the different considered parameterizations and knot placement techniques revealing in particular that with the primal formulation an equally spaced parameterization is definitively the most recommended choice and it should always be used with an approximation degree of at least p 6 although some caution must be adopted when very high jacobians and small curvatures occur the same holds for the mixed formulation with the difference that p 4 is enough to yield accurate results

two party bell inequalities derived from combinatorics via triangular elimination we establish a relation between the two party bell inequalities for two valued measurements and a high dimensional convex polytope called the cut polytope in polyhedral combinatorics using this relation we propose a method triangular elimination to derive tight bell inequalities from facets of the cut polytope this method gives two hundred million inequivalent tight bell inequalities from currently known results on the cut polytope in addition this method gives general formulae which represent families of infinitely many bell inequalities these results can be used to examine general properties of bell inequalities

isogeometric collocation for large deformation elasticity and frictional contact problems abstract isogeometric collocation methods have been recently proposed as an alternative to standard galerkin approaches as they provide a significant reduction in computational cost for higher order discretizations in this work we explore the application of isogeometric collocation to large deformation elasticity and frictional contact problems we first derive the non linear governing equations for the elasticity problem with finite deformation kinematics and provide details on their consistent linearization some numerical examples demonstrate the performance of collocation in its basic and enhanced versions differing by the enforcement of neumann boundary conditions for problems with strong singularities enhanced collocation is shown to outperform basic collocation and to lead to a spatial convergence behavior very similar to galerkin whereas for weaker or no singularities enhanced and basic collocation may give very similar results a large deformation contact formulation is subsequently developed and tested in the frictional setting where collocation confirms the excellent performance already obtained for the frictionless case finally it is shown that the contact formulation in the collocation framework passes the contact patch test to machine precision in a three dimensional setting with arbitrarily inclined non matching discretizations thus outperforming most of the available contact formulations and all those with pointwise enforcement of the contact constraints

cut polytopes boolean quadratic polytopes and nonnegative quadratic pseudo boolean functions in this paper we present a class of valid inequalities as well as a class of facets for the cut polytope of the complete graph it is shown that many of the known classes of valid inequalities e g triangle hypermetric and cycle inequalities belong to this class by specializing some of the parameters large classes of facets can be defined it is shown that each of these classes contains at least n 3n 3 facets where n 10 denotes the number of vertices and that the corresponding separation problems are polynomially solvable these results are based on a one to one correspondence established between the set of valid inequalities facets for the cut polytope of kn 1 and the set of nonnegative extremal quadratic pseudo boolean functions in n variables

a computational contact formulation based on surface potentials this work presents the theory and numerical implementation of a contact formulation based on surface potentials the new theory formulates contact based on distance dependent surface interaction potentials and distinguishes between three interaction classes point interaction short range surface interaction and long range surface interaction here the focus is placed on frictionless contact although the first class readily admits frictional sticking contact as is also shown the proposed contact theory provides a unified framework for various contact formulations ranging from numerical constraint formulations like penalty barrier cross constrained and augmented lagrange multiplier methods to physical interaction formulations like cohesive zone models as well as electrostatic gravitational and van der waals interaction apart from recovering classical penalty and barrier formulations the new theory also naturally leads to a modified penalty and barrier method the formulation also recovers classical one pass contact algorithms however the real advantages lie in a novel two pass contact algorithm denoted the two half pass contact algorithm since each pass only accounts for the contact forces acting on the slave body this implies that traction continuity is only satisfied in theory but not imposed a priori in the algorithm instead it is obtained naturally to high accuracy as is demonstrated by several 2d and 3d numerical examples these include sliding contact peeling contact and electrostatic attraction between deformable solids among the examples is a detailed comparison between the new formulation and classical one pass approaches it is further shown that the new contact formulation passes the contact patch test

avoiding shear locking for the timoshenko beam problem via isogeometric collocation methods in this work we study isogeometric collocation methods for the timoshenko beam problem considering both mixed and displacement based formulations in particular we show that locking free solutions are obtained for mixed methods independently on the approximation degrees selected for the unknown fields moreover several numerical tests are provided in order to support our theoretical results and to show the good behavior and the flexibility of isogeometric collocation methods in this context

testing the odd bicycle wheel inequalities for the bipartite subgraph polytope in their paper on facets of the bipartite subgraph polytope barahona grotschel and mahjoub pose the question is there a polynomial time algorithm testing the odd bicycle wheel inequalities this note describes such an algorithm

efficient quadrature for nurbs based isogeometric analysis abstract we initiate the study of efficient quadrature rules for nurbs based isogeometric analysis a rule of thumb emerges the half point rule indicating that optimal rules involve a number of points roughly equal to half the number of degrees of freedom or equivalently half the number of basis functions of the space under consideration the half point rule is independent of the polynomial order of the basis efficient rules require taking into account the precise smoothness of basis functions across element boundaries several rules of practical interest are obtained and a numerical procedure for determining efficient rules is presented we compare the cost of quadrature for typical situations arising in structural mechanics and fluid dynamics the new rules represent improvements over those used previously in isogeometric analysis

contact treatment in isogeometric analysis with nurbs abstract we study nurbs based isogeometric analysis of contact problems and compare with standard c 0 continuous lagrange finite elements a knot to surface kts algorithm is developed to treat the contact constraints with nurbs contact surface discretizations qualitative studies deliver satisfactory results for various finite deformation frictionless thermoelastic contact problems quantitative studies based on the hertz problem suggest the need for a relaxation of the mechanical contact constraints that appear in the standard kts approach the improved mortar based kts algorithm delivers robust and accurate results for nurbs discretizations based on numerical examples we conclude that nurbs based isogeometric analysis is a viable technology for contact problems and offers potential accuracy as well as convergence improvements over c 0 continuous finite elements

isogeometric collocation cost comparison with galerkin methods and extension to adaptive hierarchical nurbs discretizations abstract we compare isogeometric collocation with isogeometric galerkin and standard c 0 finite element methods with respect to the cost of forming the matrix and residual vector the cost of direct and iterative solvers the accuracy versus degrees of freedom and the accuracy versus computing time on this basis we show that isogeometric collocation has the potential to increase the computational efficiency of isogeometric analysis and to outperform both isogeometric galerkin and standard c 0 finite element methods when a specified level of accuracy is to be achieved with minimum computational cost we then explore an adaptive isogeometric collocation method that is based on local hierarchical refinement of nurbs basis functions and collocation points derived from the corresponding multi level greville abscissae we introduce the concept of weighted collocation that can be consistently developed from the weighted residual form and the two scale relation of b splines using weighted collocation in the transition regions between hierarchical levels we are able to reliably handle coincident collocation points that naturally occur for multi level greville abscissae the resulting method combines the favorable properties of isogeometric collocation and hierarchical refinement in terms of computational efficiency local adaptivity robustness and straightforward implementation which we illustrate by numerical examples in one two and three dimensions

enriched isogeometric collocation for two dimensional time harmonic acoustics abstract in this work the isogeometric collocation iga c is paired with two types of enrichment plane wave pw and generalized harmonic polynomial ghp functions to solve 2d problems for the helmholtz equation a parametric study is conducted and a detailed assessment of the performance of the method in a number of benchmark problems is provided three different collocation methods are tested for the non enriched formulation namely greville abscissae ga approximated cauchy galerkin acg and superconvergent sc points showing that the acg scheme is the best choice in terms of overall error convergence rate and ease of solving the linear system then the influence of the number of shape functions in the original and enriched basis location and number of collocation points and wave number over both the convergence rate as well as the condition number of the stiffness matrix are studied the numerical results show that 1 there is an improvement over the non enriched formulation 2 the improvement depends on the choice of the number and type of enrichment and 3 the pollution error is not completely alleviated with the enriched formulations

modular properties of nodal curves on k3 surfaces in this article we are going to address the following issues 1 the first is a rigidity property for pairs s c consisting of a general projective k3 surface s and a curve c obtained as the normalization of a nodal hyperplane section hat c hookrightarrow s we prove that a non trivial deformation of a pair s c induces a non trivial deformation of c 2 the second question concerns the wahl map of curves c obtained as above we prove that the wahl map of the normalization of a nodal curve contained in a general projective k3 surface is non surjective in both cases we impose upper bounds on the number of nodes of hat c

on universal severi varieties of low genus k3 surfaces we prove the irreducibility of universal severi varieties parametrizing irreducible reduced nodal hyperplane sections of primitive k3 surfaces of genus g with 3 g 11 g 10

classification of varieties with canonical curve section via gaussian maps on canonical curves the purpose of this article is to develop further a method to classify varieties x inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i having canonical curve section using gaussian map computations in a previous article we applied these techniques to classify prime fano threefolds that is fano threefolds whose picard group is generated by the hyperplane bundle in this article we extend this method and classify fano threefolds of higher index and mukai varieties i e varieties of dimension four or more with canonical curve sections first we determine when the hilbert scheme inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i of such varieties x is nonempty moreover in the case of picard number one we prove that inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i is irreducible and that the examples of fano iskovskih and mukai form a dense open subset of smooth points of inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 04i

counting rational curves on text k3 surfaces the aim of these notes is to explain the remarkable formula found by yau and zaslow y z to express the number of rational curves on a k3 surface projective k3 surfaces fall into countably many families fg g 1 a surface in fg admits a gdimensional linear system of curves of genus g a naive count of constants suggests that such a system will contain a positive number say n g of rational highly singular curves the formula is

one third integrality in the max cut problem given a graphg v e the metric polytopes g is defined by the inequalitiesx f x c f f 1 for f subseteq c f odd c cycle ofg and 0 xe 1 fore e optimization overs g provides an approximation for the max cut problem the graphg is called 1 d integral if all the vertices ofs g have their coordinates in i d 0 i d we prove that the class of 1 d integral graphs is closed under minors and we present several minimal forbidden minors for 1 3 integrality in particular we characterize the 1 3 integral graphs on seven nodes we study several operations preserving 1 d integrality in particular thek sum operation for 0 k 3 we prove that series parallel graphs are characterized by the following stronger property all vertices of the polytopes g x l x u are 1 3 integral for every choice of 1 3 integral boundsl u on the edges ofg

a discrete least squares collocation method for two dimensional nonlinear time dependent partial differential equations abstract in this paper we develop regularized discrete least squares collocation and finite volume methods for solving two dimensional nonlinear time dependent partial differential equations on irregular domains the solution is approximated using tensor product cubic spline basis functions defined on a background rectangular interpolation mesh which leads to high spatial accuracy and straightforward implementation and establishes a solid base for extending the computational framework to three dimensional problems a semi implicit time stepping method is employed to transform the nonlinear partial differential equation into a linear boundary value problem a key finding of our study is that the newly proposed mesh free finite volume method based on circular control volumes reduces to the collocation method as the radius limits to zero both methods produce a large constrained least squares problem that must be solved at each time step in the advancement of the solution we have found that regularization yields a relatively well conditioned system that can be solved accurately using qr factorization an extensive numerical investigation is performed to illustrate the effectiveness of the present methods including the application of the new method to a coupled system of time fractional partial differential equations having different fractional indices in different irregularly shaped regions of the solution domain

the variational collocation method we propose the variational collocation method for the numerical solution of partial differential equations the conceptual basis is the establishment of a direct connection between the galerkin method and the classical collocation methods with the perspective of achieving the accuracy of the former with a computational cost of one point evaluation per degree of freedom as in the latter variational collocation requires a discrete space constructed by smooth and pointwise non negative basis functions which makes the approach immediately applicable to isogeometric analysis and some meshfree methods in this paper we concentrate on isogeometric analysis and demonstrate that there exists a set of points such that collocation of the strong form at these points produces the galerkin solution exactly we provide an estimate of these points and show that applying isogeometric collocation at the estimated points completely solves the well known odd even discrepancy in the order of spatial convergence we demonstrate the potential of variational collocation with examples of linear and non linear elasticity as well as kirchhoff plates

rational curves on k3 surfaces we show that projective k3 surfaces with odd picard rank contain infinitely many rational curves our proof extends the bogomolov hassett tschinkel approach i e uses moduli spaces of stable maps and reduction to positive characteristic