linear matrix inequality representation of sets this article concerns the question which subsets of m can be represented with linear matrix inequalities lmis this gives some perspective on the scope and limitations of one of the most powerful techniques commonly used in control theory also before having much hope of representing engineering problems as lmis by automatic methods one needs a good idea of which problems can and cannot be represented by lmis little is currently known about such problems in this article we give a necessary condition that we call rigid convexity which must hold for a set m in order for to have an lmi representation rigid convexity is proved to be necessary and sufficient when m 2 this settles a question formally stated by pablo parrilo and berndt sturmfels in 15 as shown by lewis parillo and ramana 11 our main result also establishes in the case of three variables a 1958 conjecture by peter lax on hyperbolic polynomials 2006 wiley periodicals inc

fifteen node tetrahedral elements for explicit methods in nonlinear solid dynamics abstract despite the ease in meshing and benefits for modeling flexure curved shapes etc second order tetrahedral elements are not contained in typical explicit solid dynamic programs this is primarily due to the lack of both a satisfactory consistent nodal loading distribution and mass lumping technique row summation lumping for instance produces negative vertex node masses for the popular ten node serendipity tetrahedron which also has zero vertex node loads resulting from a constant pressure on an element face this has led to piecewise composites of four node tetrahedrons to represent a ten node one in explicit codes in this paper truly second order fifteen node formulations for compressible and for nearly incompressible materials are presented and evaluated in addition to producing all positive nodal loads from a uniform traction row summation mass lumping for the fifteen node element is shown to produce all positive nodal masses performance is assessed in standard benchmark problems and practical applications using various elastic and elastic plastic material models and involving very large strains deformations severe distortions and contact impact comparisons are also made with several first order elements and second order hexahedral formulations the offered elements performed satisfactorily in all examples as recently found for second order hexahedral elements it is shown that the inclusion of face and centroidal nodes is vital for robust performance with row summation lumping and high order quadrature rules are crucial with explicit methods these second order elements are shown to be viable for practical applications especially using today s parallel computers whereas the reliable performance is generally attained at significant computational expense compared with first order and brick types these elements can be more computationally competitive in flexure and have the desirable trait that they are amenable to automatic tetrahedral meshing software

p refinement and p threads abstract p type refinement leads to exponential decay of numerical errors for sufficiently smooth solutions and has been used effectively in turbulence and structural mechanics simulations in the context of spectral and hp finite element discretizations however it induces a computational cost of o p d 1 in d dimensions which is higher than lower order methods in this paper we demonstrate that by employing multi threading within mpi processes we manage to counter balance the cost increase associated with p refinement this approach reduces effectively the wall clock time and keeps it essentially constant as the polynomial order is increased while achieving exponential convergence rate since the number of threads within mpi processes can be dynamically adjusted through thread library functions the algorithm can be readily adapted for dynamic p refinement the resulting hybrid mpi threads dual level parallelism is particularly suitable for modern supercomputers consisting of symmetric multiprocessor nodes we demonstrate this approach in simulations of two three dimensional fluid dynamics problems

eigenvalues of symmetrical matrices and graphs abstract we investigate the class c of integral domains a having the property that each totally real integral element over a is an eigenvalue of a symmetric matrix over a c is shown to contain all dedekind domains and any domain in which 1 is a sum of squares in the case a z these results imply that any totally real algebraic integer is the eigenvalue of the adjacency matrix of some regular graph thus affirming a conjecture of hoffman

determinantal representations and bezoutians a major open question in convex algebraic geometry is whether all hyperbolicity cones are spectrahedral i e the solution sets of linear matrix inequalities we will use sum of squares decompositions of certain bilinear forms called bezoutians to approach this problem more precisely we show that for every smooth hyperbolic polynomial h there is another hyperbolic polynomial q such that q cdot h has a definite determinantal representation besides commutative algebra the proof relies on results from real algebraic geometry

p 2 continuous finite elements on tetrahedra with local mass matrix inversion abstract when using continuous finite element methods for unsteady simulations one must invert a global mass matrix even for an explicit time advancement scheme in this paper we develop a local mass matrix inversion procedure for p 2 c 0 tetrahedral elements and also show that unlike on triangles b t helenbrook siam j numer anal 47 2009 1304 1318 this procedure does not generalize to higher order for the p 2 basis it is demonstrated that the accuracy of the local inversion is better than any previous mass lumping scheme this allows the development of fast and low memory iterative algorithms as well as accurate explicit computations to be performed the efficiency of using the approach as an iterative relaxation scheme for solving the helmholtz equation is shown by comparing to more standard approaches such as p multigrid with jacobi relaxation lastly the local mass matrix inversion is used to march the unsteady heat equation explicitly with third order spatial accuracy

robustness of isogeometric structural discretizations under severe mesh distortion abstract this paper investigates higher order and higher continuity functions in isogeometric structural analysis under distortion of the control and physical meshes first the concepts behind isogeometric analysis are briefly reviewed the effect of perturbing control points upon the geometrical mapping is observed in the one dimensional setting the concept is extended to multiple dimensions with a series of cases where the interior of a cube is distorted in different ways and patch tests are performed to assess accuracy a circular fillet patch is introduced through the appropriate degeneration of a control mesh in order to address the problem of modeling small features in finite element models the fillet patch is tested with two problems of linear elasticity and is shown to be accurate the investigation is further extended to the setting of large deformation nearly incompressible nonlinear elasticity with two examples which make use of the recently proposed f projection method

on capacity computation for symmetric polygonal condensers abstract making use of two different analytical numerical methods for capacity computation we obtain matching to a very high precision numerical values for capacities of a wide family of planar condensers these two methods are based respectively on the use of the lauricella function bezrodnykh and vlasov 2002 bezrodnykh 2016 64 65 and riemann theta functions bogatyrev 2012 grigoriev 2013 bogatyrev and grigor ev 2017 bogatyrev and grigor ev 2018 we apply these results to benchmark the performance of numerical algorithms which are based on adaptive h p finite element method hakula et al 2011 2013 2018 and boundary integral method tsuji 1959 jaswon and symm 1977 albrecht and collatz 1980

multiparametric shell eigenvalue problems abstract the eigenproblem for thin shells of revolution under uncertainty in material parameters is discussed here the focus is on the smallest eigenpairs shells of revolution have natural eigenclusters due to symmetries moreover the eigenpairs depend on a deterministic parameter the dimensionless thickness the stochastic subspace iteration algorithms presented here are capable of resolving the smallest eigenclusters in the case of random material parameters it is possible that the eigenmodes cross in the stochastic parameter space this interesting phenomenon is demonstrated via numerical experiments finally the effect of the chosen material model on the asymptotics in relation to the deterministic parameter is shown to be negligible

characteristic polynomials of symmetric matrices over the univariate polynomial ring abstract viewing a bivariate polynomial f r x t as a family of univariate polynomials in t parametrized by real numbers x we call f real rooted if this family consists of monic polynomials with only real roots if f is the characteristic polynomial of a symmetric matrix with entries in r x it is obviously real rooted in this article the converse is established namely that every real rooted bivariate polynomial is the characteristic polynomial of a symmetric matrix over the univariate real polynomial ring as a byproduct we present a purely algebraic proof of the helton vinnikov theorem which solved the 60 year old lax conjecture on the existence of definite determinantal representation of ternary hyperbolic forms

determinants and characteristic polynomials of lie algebras abstract for an s tuple a a 1 a s of square matrices of the same size the joint determinant of a and the characteristic polynomial of a are defined by det a z det z 1 a 1 z 2 a 2 z s a s and p a z det z 0 i z 1 a 1 z 2 a 2 z s a s respectively this paper calculates determinant of the finite dimensional irreducible representations of sl 2 f which is either zero or a product of some irreducible quadratic polynomials moreover it shows that a finite dimensional lie algebra is solvable if and only if the characteristic polynomial is completely reducible

mid node admissible spaces for quadratic triangular 2d finite elements with one edge curved a robust element distortion metric based on the new concept of mid node admissible spaces for two dimensional quadratic triangular finite elements is developed the metric is based on the jacobian determinant over the entire element without requiring that it actually be computed everywhere on the element the metric is relatively inexpensive to compute especially for mildly distoted elements the metric is able to detect elementsof poor quality that other distortion metrics fall to detect it also has the ability to approve elements of good quality regardless of the extent to which they may appear geometrically distorted copyright 2001 john wiley sons ltd

interlacing families ii mixed characteristic polynomials and the kadison singer problem we use the method of interlacing polynomials introduced in our previous article to prove two theorems known to imply a positive solution to the kadison singer problem the rst is weaver s conjecture ks 2 which is known to imply kadison singer via a projection paving conjecture of akemann and anderson the second is a formulation due to casazza et al of anderson s original paving conjecture s for which we are able to compute explicit paving bounds the proof involves an analysis of the largest roots of a family of polynomials that we call the mixed characteristic polynomials of a collection of matrices

spectral element feti dp and bddc preconditioners with multi element subdomains abstract the discrete systems generated by spectral or hp version finite elements are much more ill conditioned than the ones generated by standard low order finite elements or finite differences this paper focuses on spectral elements based on gauss lobatto legendre gll quadrature and the construction of primal and dual non overlapping domain decomposition methods belonging to the family of balancing domain decomposition methods by constraints bddc and dual primal finite element tearing and interconnecting feti dp algorithms new results are presented for the spectral multi element case and also for inexact feti dp methods for spectral elements in the plane theoretical convergence estimates show that these methods have a convergence rate independent of the number of subdomains and coefficient jumps of the elliptic operator while there is only a polylogarithmic dependence on the spectral degree p and the ratio h h of subdomain and element sizes parallel numerical experiments on a linux cluster confirm these results for tests with spectral degree up to p 32 thousands of subdomains and coefficient jumps up to 8 orders of magnitude

numerical analysis of singularities in two dimensions part 1 computation of eigenpairs a numerical method is described for the computation of eigenpairs which characterize the exact solution of linear second order elliptic partial differential equations in two dimensions in the vicinity of singular points the singularities may be caused by re entrant corners and abrupt changes in boundary conditions or material properties such singularities are of great interest from the point of view of failure initiation the eigenpairs characterize the straining modes and their amplitudes quantify the amount of energy residing in particular straining modes for this reason failure theories directly or indirectly involve the eigenpairs and their amplitudes this paper addresses the problem of determining the eigenpairs numerically on the basis of the steklov formulation numerical results are presented for several cases importantly the method is applicable to three dimensional cases

three dimensional mixed explicit implicit generalized galerkin spectral element methods for high speed turbulent compressible flows abstract in high speed flows the interactions of shock waves with turbulent boundary layers are important design considerations because of the complex flowfields resulting in increased adverse pressure gradients skin friction and temperatures unsteadiness and three dimensional flowfield structure are also characteristic of shock wave turbulent boundary layer interactions such physical phenomena require sophisticated numerical schemes in the solution of governing equations the purpose of this paper therefore is to introduce an accurate and efficient approach the mixed explicit implicit generalized galerkin spectral element method mei gg sem with legendre polynomial spectral elements in which flowfield dependent implicitness parameters provide automatically adequate computational requirements for compressible and incompressible flows or high speed and low speed flows this is in contrast to the traditional approach in which all speed regime analysis requires a separate hyperbolic elliptic pressure equation for pressure correction if the flow becomes incompressible in the mei gg sem scheme mesh refinements are carried out adaptively until shock waves are resolved followed then by the adaptive increase of legendre polynomial degrees until turbulence microscales are resolved in which the traditional turbulence modeling is no longer required aimed toward direct numerical simulation in order to demonstrate the validity of the theory and numerical procedure two dimensional flat plate and compression corner high speed flows are investigated followed by a three dimensional sharp leading edged fin for swept shock wave turbulent boundary layer interactions comparisons of the present study with experimental measurements and other numerical studies show favorable agreement

on the characteristic polynomials of multiparameter pencils abstract in this paper we study the characteristic polynomial of multiparameter pencil z 1 a 1 z 2 a 2 z s a s the main theorem states that a unitary representation of a finitely generated group contains a one dimensional representation if and only if the characteristic polynomial of its generators contains a linear factor it follows that a two or three dimensional unitary representation of a finitely generated group is irreducible if and only if the characteristic polynomial of the pencil of its generators is irreducible the result is of kin to the dedekind and frobenius theorem on finite group determinant

matching experimental and three dimensional numerical models for structural vibration problems with uncertainties abstract the simulation model which examines the dynamic behavior of real structures needs to address the impact of uncertainty in both geometry and material parameters this article investigates three dimensional finite element models for structural dynamics problems with respect to both model and parameter uncertainties the parameter uncertainties are determined via laboratory measurements on several beam like samples the parameters are then considered as random variables to the finite element model for exploring the uncertainty effects on the quality of the model outputs i e natural frequencies the accuracy of the output predictions from the model is compared with the experimental results to this end the non contact experimental modal analysis is conducted to identify the natural frequency of the samples the results show a good agreement compared with experimental data furthermore it is demonstrated that geometrical uncertainties have more influence on the natural frequencies compared to material parameters and material uncertainties are about two times higher than geometrical uncertainties this gives valuable insights for improving the finite element model due to various parameter ranges required in a modeling process involving uncertainty

locking free mixed hp finite element methods for curvilinear domains mixed hp finite element methods fems are robust with respect to poisson ratio locking when straight sided quadrilateral elements are used for the p version which uses fixed and generally large elements curved elements are necessary to represent curved domains accurately we show that mixed hp curved quadrilateral elements retain the robustness of their straight sided counterparts provided degenerate elements are avoided we also show that hp version triangles are robust we use these concepts to develop a locking free fem for geometrically nonlinear elasticity

eigenvalues of symmetric matrices over integral domains abstract given an integral domain a we consider the set of all integral elements over a that can occur as an eigenvalue of a symmetric matrix over a we give a sufficient criterion for being such an element in the case where a is the ring of integers of an algebraic number field this sufficient criterion is also necessary and we show that the size of matrices grows linearly in the degree of the element the latter result settles a questions raised by bass estes and guralnick

stability criteria for matrices in this paper we study the stability of matrices using technologies of special matrices analysis several simple criteria for the stability for the matrix a aij cn n which is not diagonally dominant are obtained by aid of simple algebraic conditions of aij only