a critical p biharmonic system with negative exponents abstract in this paper we are devoted to the study of critical p biharmonic system with a parameter which involves strongly coupled critical nonlinearities and negative exponents we prove that there exists a positive constant such that the above problem admits at least two solutions if 0

on a liouville type equation with sign changing weight in this paper we study the existence nonexistence and multiplicity of non negative solutions for the family of problems u a x e f x u u h 0 where is a bounded domain in r2 and 0 is a parameter the coefficient a x is allowed to change sign the techniques used in the proofs are a combination of upper and lower solutions the trudingermoser inequality and variational methods note that when f x u 0 the equation is of liouville type

infinitely many nodal solutions for nonlinear elliptic equations with combined nonlinearities and lack of symmetry abstract we present a new approach to studying a parametric nonlinear dirichlet problem driven by a nonhomogeneous differential operator and with a reaction which is concave i e p 1 sublinear near zero and convex i e p 1 superlinear near the reaction term is not assumed to be odd we show that for all small values of the parameter 0 the problem has infinitely many nodal solutions

adaptivity in space and time for reaction diffusion systems in electrocardiology the paper introduces and studies numerical methods that are fully adaptive in both three dimensional 3d space and time to challenging multiscale cardiac reaction diffusion models in these methods temporal adaptivity comes via stepsize control in function space oriented linearly implicit time integration while spatial adaptivity is realized within multilevel finite element methods controlled by a posteriori local error estimators in contrast to other recent adaptivity approaches to cardiac modeling that discretize first in space and then in time so called method of lines our method discretizes first in time and then in space so called rothe method an approach that has already proven to be highly efficient in a number of challenging multiscale problems in science and technology kardos code library with this method the evolution of a complete heartbeat from the excitation to the recovery phase is simulated both in the frame of the anisotropic monodomain models and in the more realistic anisotropic bidomain models coupled with either a variant of the simple fitzhugh nagumo model or the more complex phase i luo rudy ionic model the numerical results exhibit a rather satisfactory performance of our adaptive method for complex cardiac reaction diffusion models on 3d domains up to moderate sizes in particular the method accurately resolves the evolution of the intra and extracellular potentials gating variables and ion concentrations during the excitation plateau and recovery phases

local superlinearity and sublinearity for the p laplacian we study the existence nonexistence and multiplicity of positive solutions for a family of problems updelta p u f lambda x u u in w 0 1 p upomega where is a bounded domain in mathbb r n n p and 0 is a parameter the family we consider includes the well known nonlinearities of ambrosetti brezis cerami type in a more general form namely lambda a x u q b x u r where 0 leqslant q p 1 r leqslant p 1 here the coefficient a x is assumed to be nonnegative but b x is allowed to change sign even in the critical case preliminary results of independent interest include the extension to the p laplacian context of the brezis nirenberg result on local minimization in w 0 1 p text and c 0 1 a c 1 alpha estimate for equations of the form updelta p u h x u with h of critical growth a strong comparison result for the p laplacian and a variational approach to the method of upper lower solutions for the p laplacian

combined effects of concave and convex nonlinearities in elliptic equation on rn abstract in this paper we consider the existence of two positive solutions of the elliptic equation on r n with concave and convex nonlinearities

limits as p of p laplacian concave convex problems abstract we study the behavior as p of the sequel of positive weak solutions of the concave convex problem p div u p 2 u u q p u r p in u 0 in u 0 on where r n is a bounded domain 0 and the exponents q r satisfy lim p q p p 1 q lim p r p p 1 r with 0 q 1 r we characterize any positive uniform limit of a sequence of weak solutions of p as a viscosity solution of min u max u q u r u 0 in notice that the limit process decouples the nonlinearity we obtain existence non existence and global multiplicity of positive viscosity solutions of the limit problem in terms of the parameter

existence and multiplicity of solutions for a nonlocal problem with critical sobolev exponent abstract in this work we are interested in considering the following nonlocal problem a b u 2 d x u u 2 2 u u q 2 u x u 0 x where r n n 4 is a smooth bounded domain a 0 b 0 1 q 2 0 and 2 2 n n 2 is the critical sobolev exponent by using the variational method and the critical point theorem some existence and multiplicity results are obtained

on semilinear elliptic equations involving concave convex nonlinearities and sign changing weight function abstract in this paper we study the combined effect of concave and convex nonlinearities on the number of positive solutions for semilinear elliptic equations with a sign changing weight function with the help of the nehari manifold we prove that there are at least two positive solutions for eq e f in bounded domains

estimation of reciprocal residual flexibility from experimental modal data a technique is presented for estimating the residual flexibility between nonexcited structural degrees of freedom from experimental structural vibration data using this method one can include the residual flexibility estimated from modal measurements in the computation of measured flexibility for experiments with incomplete reciprocity i e when the response and excitation measurement sensors are not fully collocated the method can also be used to estimate the unknown entries in the residual flexibility matrix for experimental component mode synthesis when excitations are not provided at all interface degrees of freedom a general solution is presented that contains an unknown positive semidefinite contribution the general solution satisfies modal orthogonality in the limit that all of the structural degrees of freedom are instrumented and when the positive semidefinite contribution lies in a nullspace defined by the stiffness matrix and the modal flexibility with a limited number of measurements modal orthogonality is shown to be satisfied to the extent that the measured modes are preserved by static condensation a rank deficient solution is presented that allows the residual to be used in the computation of the flexibility matrix without further modeling assumptions numerical and experimental results that demonstrate the application of the method to both flexibility matrix convergence and experimental component mode synthesis are presented

bpx type preconditioners for second and fourth order elliptic problems on the sphere we develop two bramble pasciak xu type preconditioners for second resp fourth order elliptic problems on the surface of the two sphere to discretize the second order problem we construct c 0 linear elements on the sphere and for the fourth order problem we construct c 1 finite elements of powell sabin type on the sphere the main reason these bpx preconditioners work depends on this particular choice of basis we prove optimality and provide numerical examples furthermore we numerically compare the bpx preconditioners with the suboptimal hierarchical basis preconditioners

structural damage assessment using a generalized minimum rank perturbation theory recently the authors proposed computationally attractive algorithms to determine the location and extent of structural damage for undamped structures assuming damage results in a localized change in stiffness properties the algorithms make use of a finite element model and a subset of measured eigenvalues and eigenvectors the developed theories approach the damage location and extent problem in a decoupled fashion first a theory is developed to determine the location of structural damage with location determined a damage extent theory is then developed the damage extent algorithm is a minimum rank perturbation which is consistent with the effects of many classes of structural damage on a finite element model in this work the concept of the minimum rank perturbation theory mrpt is adopted to determine the damage extent on the mass properties of an undamped structure in addition the mrpt is extended to the case of proportionall y damped structures for proportionally damped structures the mrpt is used to find the damage extent in any two of the three structural property matrices mass damping or stiffness finally illustrative case studies using both numerical and actual experimental data are presented he advent of the space shuttle has prompted considerable attention to the design and control of large space structures due to the large size and complexity of envisioned structures as well as the use of advanced materials to reduce structural weight it may become necessary to develop a structural health monitoring system to detect and locate structural damage as it occurs from experience gained in the machinery health monitoring field one would expect the vibration signature of the structure either frequency response functions and or modal parameters to provide useful information in determining the location and extent of structural damage assume that a refined finite element model fem of the structure has been developed before damage has occurred by refined we mean that the measured and analytical modal properties are in agreement next assume that at a later date some form of structural damage has occurred if significant the damage will result in a change in the structures modal parameters the question is can the discrepancy between the original fem modal properties and postdamage modal properties be used to ascertain structural damage most prior work in damage detection has used the general framework of fem refinement system identification in the development of damage assessment algorithms the motivation behind the development of fem refinement techniques is based on the need to validate engineering fems before their acceptance as the basis for final design analysis the standard problem has been to seek a refined fem that is as close to the original fem and whose modal properties are in agreement with those that are measured subject to various constraints such as symmetry and sparsity preservation a considerable amount of work in this area has been

nonnegative solutions of elliptic problems with sublinear indefinite nonlinearity abstract we are concerned with existence nonexistence and multiplicity of nonnegative solutions for the elliptic problem u a x u q b x u p in u 0 on where is a bounded domain in r n r 0 q 1 p 2 1 and a b are bounded functions with b x 0 and a x changes its sign

positive solutions of superlinear elliptic equations abstract in this paper we study the existence of two positive solutions of superlinear elliptic equations without assuming the conditions which have been used in the literature to deduce either the p s condition or a priori bounds of positive solutions the first solution is proved as the minimal positive solution while the second one is obtained as the limit of a gradient flow whose starting point is properly chosen the dependence of the minimal solution upon a parameter is also considered

exact multiplicity and bifurcation diagrams of positive solutions of a one dimensional multiparameter prescribed mean curvature problem abstract we study the exact multiplicity and bifurcation diagrams of positive solutions u c 2 l l c l l of the one dimensional multiparameter prescribed mean curvature problem u x 1 u x 2 u p u q l x l u l u l 0 where 0 is a bifurcation parameter l 0 the radius of the one dimensional ball l l is an evolution parameter and 0 p q are two constants we prove that the problem has at most two positive solutions for any 0 p q and l 0 in addition if 0 p q q p p 1 2 p 1 we give a classification of totally three qualitatively different bifurcation diagrams on the u plane for any l 0 for any fixed p 0 and q q p p 2 2 2 p 3 we prove that there exist positive l l such that the bifurcation diagrams on the u plane are individually qualitatively different for the cases i 0 l l or l l ii l l iii l l l

correcting finite element models using a symmetric eigenstructure assignment technique improvement of structural models by incorporating measured structural modal parameters is approached from a controls aspect the approach is developed for linear structures that exhibit nonproportional damping residual damping and stiffness matrices are determined such that the improved analytical model eigenstructure matches more closely that obtained experimentally

updating finite element dynamic models using an element by element sensitivity methodology a sensitivity based methodology for improving the finite element model of a given structure using test modal data and a few sensors is presented the proposed method searches for both the location and sources of the mass and stiffness errors and does not interfere with the theory behind the finite element model while correcting these errors the updating algorithm is derived from the unconstrained minimization of the squared 2 norms of the modal dynamic residuals via an iterative two step staggered procedure at each iteration the measured mode shapes are first expanded assuming that the model is error free then the model parameters are corrected assuming that the expanded mode shapes are exact the numerical algorithm is implemented in an element by element fashion and is capable of zooming on the detected error locations several simulation examples which demonstrate the potential of the proposed methodology are discussed i introduction e ven though the field of finite element analysis of structural dynamic problems has witnessed tremendous progress in the last two decades greater confidence is still placed in experimental data therefore the finite element model of a given structure is often updated to reflect the results of a particular experiment usually the computed eigenmodes are compared with the measured frequencies and mode shapes if both sets are not in agreement the finite element model is refined via a two step updating procedure which 1 locates the errors and 2 corrects them clearly the first step is the most challenging of the two once the location of the errors is known it is relatively easy to correct them especially if the error sources can be identified however locating and identifying these errors can be a difficult task for the following reasons in general only a few experimental modes are available and these may be contaminated with random and systematic measuring errors moreover only a subset of the degrees of freedom dof in the finite element model can be monitored and as for practical and economical reasons only a few sensors can be utilized

method for determining minimum order mass and stiffness matrices from modal test data a new method for determining mass and stiffness matrices from modal test data is presented the method builds on the identified modes and the mass normalized mode shapes at the sensor locations and is not limited by either the number of driving points or measurement points and so it is applicable to most general test settings a mixed coordinate basis is defined for the mass and stiffness matrices which is analogous to the craig bampton component mode synthesis method for finite element models the resultant mass and stiffness are of minimal order necessary to span the measured modes and the resulting generalized coordinates provide an objective basis for the test derived matrices to be used as if they are component mode synthesized finite element matrices inclusion of rigid body modes and the relationship of the new method to traditional physical parameter computations based on mobility curves is considered examples of the method as applied to numerical and experimental data are provided

semi implicit time discretization schemes for the bidomain model the bidomain model is a system of partial differential equations used to model the propagation of electrical potential waves in the myocardium it is composed of coupled parabolic and elliptic partial differential equations as well as at least one ordinary differential equation to model the ion activity through the cardiac cell membranes the purpose of this paper is to propose and analyze several implicit semi implicit and explicit time stepping methods to solve that model in particular to avoid the expensive resolution of a nonlinear system through the newton raphson method we identify necessary stability conditions on the time step delta t for the proposed methods through a theoretical analysis based on energy estimates we next compare the methods for one and two dimensional test cases in terms of both stability and accuracy of the numerical solutions the theoretical stability conditions are seen to be consistent with those observed in practice our analysis allows us to recommend using either the crank nicolson adams bashforth method or the second order semi implicit backward differention method these semi implicit methods produce a good numerical solution unlike the explicit methods their stability does not depend on the spatial grid size and unlike the implicit methods they do not require the resolution of a system of nonlinear equations

an experimentally validated damage detection theory in smart structures abstract a method for the non destructive detection and location of damage using parameterized partial differential equations and galerkin approximation techniques is presented damages in a structure cause changes in the physical coefficients of mass density elastic modulus and damping coefficients this paper examines the use of beam like structure with piezoceramic sensors and actuators to perform identification of those physical parameters and hence to detect the damage the method casts the inverse problem as an optimization problem the iterative method is based on enhanced least square error minimization experimental results are presented from tests on cantilevered aluminum beams damaged at different locations and with damage of different dimensions it is demonstrated that the method can sense the presence of damage and locate and characterize the damage to a satisfactory precision

determination of an optimal regularization factor in system identification with tikhonov regularization for linear elastic continua this paper presents a geometric mean scheme gms to determine an optimal regularization factor for tikhonov regularization technique in the system identification problems of linear elastic continua the characteristics of non linear inverse problems and the role of the regularization are investigated by the singular value decomposition of a sensitivity matrix of responses it is shown that the regularization results in a solution of a generalized average between the a priori estimates and the a posteriori solution based on this observation the optimal regularization factor is defined as the geometric mean between the maximum singular value and the minimum singular value of the sensitivity matrix of responses the validity of the gms is demonstrated through two numerical examples with measurement errors and modelling errors copyright 2001 john wiley sons ltd

combined effects in nonlinear problems arising in the study ofanisotropic continuous media the paper deals with the study of a lane emden fowler equation with dirichlet boundary condition and variable potential functions the analysis developed in this paper combines monotonicity methods with variational arguments

on a dirichlet problem with p q laplacian and parametric concave convex nonlinearity abstract a homogeneous dirichlet problem with p q laplace differential operator and reaction given by a parametric p convex term plus a q concave one is investigated a bifurcation type result describing changes in the set of positive solutions as the parameter 0 varies is proven since for every admissible the problem has a smallest positive solution u both monotonicity and continuity of the map u are studied

elliptic problems with lack of compactness via a new fixed point theorem abstract this paper provides a new fixed point theorem for increasing self mappings g b b of a closed ball b x where x is a banach semilattice which is reflexive or has a weakly fully regular order cone x by means of this fixed point theorem we are able to establish existence results of elliptic problems with lack of compactness

a multilevel hybrid newton krylov schwarz method for the bidomain model of electrocardiology abstract a multilevel hybrid newton krylov schwarz nks method is constructed and studied numerically for implicit time discretizations of the bidomain reaction diffusion system in three dimensions this model describes the bioelectrical activity of the heart by coupling two degenerate parabolic equations with a stiff system of ordinary differential equations the nks bidomain solver employs an outer inexact newton iteration to solve the nonlinear finite element system originating at each time step of the implicit discretization the jacobian update during the newton iteration is solved by a krylov method employing a multilevel hybrid overlapping schwarz preconditioner additive within the levels and multiplicative among the levels several parallel tests on linux clusters are performed showing that the convergence of the method is independent of the number of subdomains scalability the discretization parameters and the number of levels optimality

existence of solutions for perturbed fractional p laplacian equations abstract the purpose of this paper is to investigate the existence of weak solutions for a perturbed nonlinear elliptic equation driven by the fractional p laplacian operator as follows p s u v x u p 2 u a x u r 2 u b x u q 2 u in r n where is a real parameter p s is the fractional p laplacian operator with 0 s 1 p p r min q p s and v a b r n 0 are three positive weights using variational methods we obtain nonexistence and multiplicity results for the above mentioned equations depending on and according to the integrability properties of the ratio a q p b r p our results extend the previous work of autuori and pucci 2013 5 to the fractional p laplacian setting furthermore we weaken one of the conditions used in their paper hence the results of this paper are new even in the fractional laplacian case

critical point theory on partially ordered hilbert spaces we develop some abstract critical point theory in order to prove that boundary value problems like the model problem formula on a bounded domain rn 2 p 2n n 2 have infinitely many sign changing solutions uk k n which are not comparable that is uk ul and uk ul change sign for k l we also show that there are no subsolutions u such that u uk for some k and u is positive somewhere a corresponding nonexistence result applies to supersolutions related results on the existence of sign changing solutions hold for other classes of nonlinearities

operator splitting and adaptive mesh refinement for the luo rudy i model we apply second order operator splitting to the luo rudy i model for electrical wave propagation in the heart the purpose of the operator splitting is to separate the nonlinear but local reaction computations from the linear but globally coupled diffusion computations this approach allows us to use local nonlinear iterations for the stiff nonlinear reactions and to solve global linear systems for the implicit treatment of diffusion for computational efficiency we use dynamically adaptive mesh refinement amr involving hierarchies of unions of grid patches on distinct levels of refinement the linear system for the discretization of the diffusion on the composite amr grid is formulated via standard conforming finite elements on unions grid patches within a level of refinement and aligned mortar elements along interfaces between levels of refinement the linear systems are solved iteratively by preconditioned conjugate gradients our preconditioner uses multiplicative domain decomposition between levels of refinement the smoother involves algebraic additive domain decomposition between patches within a level of refinement and gauss seidel iteration within grid patches numerical results are presented in 1d and 2d including spiral waves

second order structural identification procedure via state space based system identification a theory is presented for transforming the system theory based realization models into the corresponding physical coordinate based structural models the theory has been implemented into a computational procedure and applied to several example problems the results show that the present transformation theory yields an objective model basis possessing a unique set of structural parameters from an infinite set of equivalent system realization models for proportionally damped systems the transformation directly and systematically yields the normal modes and modal damping when nonproportional damping is present the relative magnitude and phase of the damped mode shapes are separately characterized and a corrective transformation is then employed to capture the undamped normal modes and nondiagonal modal damping matrix namics equations has hampered the application and acceptance of the system theory based structural identification techniques and can eventually curtail the progress of hybrid experimental analyti cal modeling and design efforts the present paper offers a theory for transforming the system theory based realization models into the corresponding physical coordinate based structural models since a key idea employed in the development of the present theory is an objective common basis normalization it is designated as a common basis normal ized structural identification cbsi procedure the resultant model is unique for a given sequence of markov parameters which in turn are uniquely determined for a linear structure with given inputs and outputs therefore the realized model after transformation has a one to one correspondence with the physical parameters of the system and can either yield data for finite ele ment model correlation or alternatively for direct calculation of mass stiffness and damping matrices of the original structure specifically we begin with the so called mcmillan transformation employed by longman and juang9 and show that the mcmillan transformation does not in general yield the desired structural nor mal modes to arrive at an objective transformation basis of gener alized coordinates we invoke two invariance properties the first is the output invariance property viz the outputs are invariant with respect to any choice of generalized coordinates the second is the normal mode identity viz for proportionally damped cases the mode shapes for the displacement velocity and acceleration vectors are the same there are several byproducts that the present theory provides primarily due to the common basis normalization employed in the theory first the present transformation theory allows the integra tion of different realized models with varying actuator and sensor locations if they arise from the same structure second each sensor and actuator or groups of sensors and actuators can be processed in parallel and combined concurrently or sequentially for the con struction of a global model third it can facilitate a decentralized real time control implementation from a structural dynamics point of view the transformation to an objective basis provides a state space model coinciding with the canonical form of the second order equations of motion thus extracting the classical real valued parameters of interest to modal testing mass normalized normal modes and the general modal damping matrix while maintaining the system equivalence properties of the state space form

a scalable newton krylov schwarz method for the bidomain reaction diffusion system a novel two level newton krylov schwarz nks solver is constructed and analyzed for implicit time discretizations of the bidomain reaction diffusion system in three dimensions this multiscale system describes the bioelectrical activity of the heart by coupling two degenerate parabolic equations with several ordinary differential equations at each point in space together with a finite element discretization in space the proposed nks bidomain solver employs an outer inexact newton iteration to solve the nonlinear finite element system originating at each time step of the decoupled implicit discretization the jacobian update during the newton iteration is solved by a krylov method employing a two level overlapping schwarz preconditioner a convergence rate estimate is proved for the resulting preconditioned operator showing that its condition number is independent of the number of subdomains scalability and bounded by the ratio of the subdomains characteristic size and the overlap size this theoretical result is confirmed by several parallel simulations employing up to more than 2000 processors for scaled and standard speedup tests in three dimensions the results show the scalability of the proposed nks bidomain solver in terms of both nonlinear and linear iterations in both cartesian slabs and ellipsoidal cardiac domains

the nehari manifold for a semilinear elliptic system involving sign changing weight functions abstract in this paper we study the combined effect of concave and convex nonlinearities on the number of solutions for a semilinear elliptic system e with sign changing weight function with the help of the nehari manifold we prove that the system has at least two nontrivial nonnegative solutions when the pair of the parameters belongs to a certain subset of r 2

model update using modal contribution to static flexibility error a new technique for the parametric correction of full order analytical stiffness matrices from reduced order modal measurements is presented the proposed algorithm corrects model parameters by minimizing a matrix residual formed at measurement degrees of freedom only the residual is the difference between the measured modal contribution to the static flexibility matrix and the corresponding analytical modal contribution to the static flexibility matrix analytical expressions are developed for the flexibility matrix error residual gradient in terms of modal sensitivities found via nelson s method by utilizing a flexibility based matrix error residual the effect of poorly modeled inertia properties on stiffness parameter estimates is greatly reduced unlike technlques that utilize only static data this technique is suitable for use on unconstrained structures in addition the method avoids the problems of mode selection determining modal correspondence and eigenvector expansion or model reduction numerical simulation results are presented for kabe s model as well as a finite element model of a welded frame structure

stiffness matrix adjustment using mode data a procedure is introduced that uses in addition to mode data structural connectivity information to optimally adjust deficient stiffness matrices the adjustments performed are such that the percentage change to each stiffness coefficient is minimized the physical configuration of the analytical model is preserved and the adjusted model will exactly reproduce the modes used in the identification the theoretical development is presented and the procedure is demonstrated by numerical simulation of a test problem

multilevel additive schwarz preconditioners for the bidomain reaction diffusion system multilevel additive schwarz methods are analyzed and studied numerically for the anisotropic cardiac bidomain model in three dimensions this is the most complete model to date of the bioelectrical activity of the heart tissue consisting of a degenerate parabolic system of nonlinear reaction diffusion equations coupled with a stiff system of several ordinary differential equations describing the ionic currents through the cellular membrane due to the presence of very different scales in both space and time the numerical discretization of this system by finite elements in space and semi implicit methods in time produces very ill conditioned linear systems that must be solved at each time step the proposed multilevel algorithm employs a hierarchy of nested meshes with overlapping schwarz preconditioners on each level and is fully additive hence parallel within and among levels convergence estimates are proved for the resulting multilevel algorithm showing that its convergence rate is independent of the number of subdomains scalability of the mesh sizes of each level and of the number of levels optimality several parallel tests on a linux cluster confirm the scalability and optimality of the method as well as its parallel efficiency on both cartesian and deformed domains in three dimensions

multiple solutions to p laplacian problems with concave nonlinearities in this paper we study the p laplacian type elliptic problems with concave nonlinearities using some asymptotic behavior of f at zero and infinity three nontrivial solutions are established

compact sobolev embeddings and positive solutions to a quasilinear equation with mixed nonlinearities abstract in this paper we study the existence and multiplicity results of nontrivial positive solutions to the following quasilinear elliptic equation on r n when n 2 p u k x u r 1 v x u q 1 here k x v x 0 are suitable potentials 1 p r q and 0 is a parameter to study this problem some compact embedding results regarding m v q p r n l k r r n are proved that unify and extend some recent results of the author 12 13 14

multiplicity of positive solutions for some quasilinear elliptic equation in rnwith critical sobolev exponent consider the equation pu g x u p 2u f x u p 2u 1 in rn where 1 0 be the principal eigenvalue of pu g x u p 2uinrn rng x u p 0 2 withu 1 0 the associated eigenfunction we prove i equation 1 has at least one positive solution if 0 1 ii suppose rn f x u 1 p 1such that 1 has at least one positive solution for 1 0 moreover ifp 2 there exists 0 1such that 1 has at least two positive solutions for 1 0

a hybrid multilevel schwarz method for the bidomain model abstract a hybrid multilevel schwarz method is studied numerically for the anisotropic bidomain model in both two and three dimensions this multiscale system models the electrical activity of the heart and it consists of two degenerate parabolic non linear reaction diffusion equations coupled with a stiff system of ordinary differential equations the numerical discretization of the whole system by finite elements in space and semi implicit methods in time generates ill conditioned linear systems that must be solved at each time step the multilevel algorithm studied employs a hierarchy of nested meshes with overlapping schwarz preconditioners on each level and is additive within the levels and multiplicative among the levels we perform several parallel tests on two linux clusters showing that the convergence of the method is independent of the number of subdomains scalability the discretization parameters and the number of levels optimality moreover the comparison with the traditional block jacobi ilu parallel preconditioner and the v cycle multigrid parallel preconditioner shows that on a whole heart cycle simulation the proposed method attains the best performances in terms of cpu times

exact multiplicity and numerical computation of solutions for two classes of non autonomous problems with concave convex nonlinearities abstract we establish the exact multiplicity of positive solutions and the global solution structure for two classes of problems on circular domains the first class involves non autonomous concave convex problems on a ball in r n and the second one deals with concave convex problems on a thin annulus in r n we illustrate our results by numerical computations using a novel algorithm which involves continuation in a global parameter

a model based block triangular preconditioner for the bidomain system in electrocardiology we introduce a preconditioner for the solution of the bidomain system governing the propagation of action potentials in the myocardial tissue the bidomain model is a degenerate parabolic set of nonlinear reaction diffusion equations the nonlinear term describes the ion flux at the cellular level the degenerate nature of the problem results in a severe ill conditioning of its discretization our preconditioning strategy is based on a suitable adaptation of the monodomain model a simplified version of the bidomain one which is by far simpler to solve nevertheless is unable to capture significant features of the action potential propagation the monodomain preconditioner application to a non symmetric formulation of the bidomain system results at the algebraic level in a lower block triangular preconditioner we prove optimality of the preconditioner with respect to the mesh size and corroborate our theoretical results with 3d numerical simulations both on idealized and real ventricle geometries

a flexibility based continuum damage identification approach the present work presents a flexibility based continuum damage identification approach here it is considered that the current integrity state of a structure is described by means of a continuum damage model this model enables one to parameterize the stiffness matrix of the finite element fe model of the system such that in an updating process properties such as symmetry and sparsity of the original fe model are naturally preserved the damage identification is accomplished by minimizing the frobenius norm of the difference between the experimental and analytical flexibility matrix only related to the measured degrees of freedom dof of the structure by utilizing an error only related to the measured dof one avoids the need for model expansion or model reduction difficulties commonly present in the model updating process the modal information required for the experimental flexibility matrix calculation are obtained by means of eigensystem realization algorithm era and common based structure identification cbsi the assessment of the present approach has been performed by means of simulations on a beam like structure considering typical shortcomings occurring in real applications such as small number of sensors different levels of signal to noise ratio and limited spectral information it has also been analyzed the use of a regularization term in the error function to be minimized in order to improve the performance of the proposed method

combined effects of concave and convex nonlinearities in some elliptic problems abstract this paper deals with a class of semilinear elliptic dirichlet boundary value problems where the combined effects of a sublinear and a superlinear term allow us to establish some existence and multiplicity results

existence and asymptotic behavior of ground states for quasilinear singular equations involving hardy sobolev exponents abstract we study the existence and decaying rate of solutions for the quasilinear problem p u x f u x g u in r n u 0 in r n u x 0 where p stands for the p laplacian operator 1 p n r n 0 is continuous and not identically zero 0 is a parameter x is the euclidean norm of x 0 p f g 0 0 are continuous and nondecreasing f has sublinear growth and the hardy sobolev exponent p p n n p bounds the growth of g we deal with variational methods and the lower and upper solutions technique

a nonlinear elliptic equation with critical exponent estimates for extremal values abstract in this work we study the values of p q the supremum of the set of related to existence and multiplicity of positive solutions of semilinear elliptic equations involving a critical sobolev exponent and establish uniform lower bounds for

four positive solutions for a semilinear elliptic equation involving concave and convex nonlinearities abstract in this paper we study the combined effect of concave and convex nonlinearities on the number of positive solutions for a semilinear elliptic equation with the help of the nehari manifold and the center mass function we prove that there are at least four positive solutions for a semilinear elliptic equation in a finite strip with a hole

normalized ground states for the nls equation with combined nonlinearities abstract we study existence and properties of ground states for the nonlinear schrodinger equation with combined power nonlinearities u u u q 2 u u p 2 u in r n n 1 having prescribed mass r n u 2 a 2 under different assumptions on q p a 0 and r we prove several existence and stability instability results in particular we consider cases when 2 q 2 4 n p 2 q p i e the two nonlinearities have different character with respect to the l 2 critical exponent these cases present substantial differences with respect to purely subcritical or supercritical situations which were already studied in the literature we also give new criteria for global existence and finite time blow up in the associated dispersive equation

efficient algebraic solution of reaction diffusion systems for the cardiac excitation process in this paper we deal with the problem of solving the large algebraic linear system arising in the numerical solution of a reaction diffusion r d system associated with myocardial excitation process modeling we show that an ad hoc preconditioning technique can be devised so as to efficiently and simultaneously handle the differential equations of the r d system with no additional memory requirements two different formulations are commonly considered for the theoretical and numerical analyses respectively we observe that the formulation employed for the theoretical analysis of the problem actually yields the best numerical performance when compared with the usual numerical scheme

multiplicity of positive solutions for p laplace equation with superlinear type nonlinearity abstract in this work we consider the problem p r p u u p u q u 0 in b r 0 u 0 on b r 0 where p 2 p np n p and b r 0 is the open ball of radius r centered about the origin in r n n 3 under the condition that p 1 q p 2 n p 1 we show that there exists r 0 such that p r admits atleast two solutions for all r r and no solution if r this extends the results of nonlinear anal tma 10 8 1986 755 to the case of p laplace equation

on elliptic problems involving critical hardy sobolev exponents and sign changing function abstract in this paper we deal with the existence and nonexistence of nonnegative nontrivial weak solutions for a class of degenerate quasilinear elliptic problems with weights and nonlinearity involving the critical hardy sobolev exponent and a sign changing function some existence results are obtained by splitting the nehari manifold and by exploring some properties of the best hardy sobolev constant together with an approach developed by brezis and nirenberg

parallel multilevel schwarz and block preconditioners for the bidomain parabolic parabolic and parabolic elliptic formulations the aim of this work is to develop parallel multilevel and block preconditioners for the bidomain model of electrocardiology the bidomain model describes the electrical activity of the heart tissue and consists of a system of two parabolic nonlinear partial differential equations pdes of reaction diffusion type pp formulation or alternatively of a system of a parabolic nonlinear pde and an elliptic linear pde pe formulation in both formulations the pdes are coupled with a system of ordinary differential equations modeling the cellular membrane ionic currents the first goal of the present study is to construct analyze and numerically test a multilevel additive schwarz preconditioner for the pe formulation of the bidomain model extending previous results obtained for the pp formulation optimal convergence rate estimates are established and confirmed by 3d numerical test on linux clusters the second goal of the present study is to analyze the scalability of multilevel schwarz block diagonal and block factorized preconditioners for both pp and pe formulations of the bidomain model and to compare them with multilevel schwarz coupled preconditioners the 3d parallel numerical tests show that block preconditioners for the pp formulation are not scalable while they are scalable for the pe formulation but less efficient than the coupled preconditioners

a variational approach to discontinuous problems with critical sobolev exponents abstract we employ variational techniques to study the existence and multiplicity of positive solutions of semilinear equations of the form u h x h u a uq u2 1 in rn where a 0 are parameters h x is both nonnegative and integrable on rn h is the heaviside function 2 is the critical sobolev exponent and 0 q

nondegeneracy and uniqueness of positive solutions for robin problem of second order ordinary differential equations and its applications abstract this article studies positive solutions of robin problem for semi linear second order ordinary differential equations nondegeneracy and uniqueness results are proven for homogeneous differential equations necessary and sufficient conditions for the existence of one or two positive solutions for inhomogeneous differential equations or differential equations with concave convex nonlinearities are obtained by making use of the nondegeneracy and uniqueness results for positive solutions of homogeneous differential equations

damage assessment of structures using modal test data abstract system health monitoring of structures is important not only for conducting safe operation but also maintaining system performance in this paper two identification algorithms for assessing structural damages using the modal test data have been developed an important characteristic in the present approaches is that the employment of the global numerical models e g fem model and some important information e g young s modulus of structures are avoided to a great extent as the first step of the damage identification two algorithms for the detection of damage location are proposed which are similar in concept to the subspace rotation algorithm or best achievable eigenvector technique furthermore a quadratic programming model is set up for the two approaches to predict the damage extent to demonstrate the capability of the proposed approaches an example of a 10 bay planar truss structure is employed for checking the present approaches numerically furthermore the experimental data from the vibration test of a beam with two fixed ends are used directly in the present approaches the final results show that the present techniques perform quite well in spite of the little structural information and measurement inaccuracies

radial solutions of dirichlet problems with concave convex nonlinearities abstract we prove the existence of a double infinite sequence of radial solutions for a dirichlet concave convex problem associated with an elliptic equation in a ball of r n we are interested in relaxing the classical positivity condition on the weights by allowing the weights to vanish the idea is to develop a topological method and to use the concept of rotation number the solutions are characterized by their nodal properties

finite difference approximate solutions for a mixed sub superlinear equation in ben mabrouk and ben mohamed 2006 a ben mabrouk m l ben mohamed on some critical and slightly super critical sub superlinear equations far east j appl math 23 1 2006 73 90 special volume of pdes a ben mabrouk m l ben mohamed nodal solutions for some nonlinear elliptic equations appl math comp in press doi 10 1016 j amc 2006 08 003 it has proved some theoretical results dealing with some boundary value problem du f u 0 in b and u 0 on b where b is a domain in r d with smooth boundary in the present paper we perform a difference scheme method to approximate the solution of a nonlinear evolutionary problem associated to the elliptic problem studied in ben mabrouk and ben mohamed 2006 we give at the end some numerical implementations based on a mixed sublinear superlinear term of the form f u u p email protected u q 1u

existence of nonnegative solutions for quasilinear elliptic equations with indefinite critical nonlinearities abstract we study the quasilinear elliptic equation div u u a x f u b x g u in with dirichlet boundary condition u 0 on where is a bounded domain in r n a x b x are sign changing continuous functions and g u has critical growth at infinity with respect to the principal part a nonnegative nontrivial solution is given under appropriate growth conditions on f u g u at 0 and

existence results for quasilinear elliptic exterior problems involving convection term and nonlinear robin boundary conditions abstract in this paper the authors establish the existence of solutions for a class of elliptic exterior problems involving convection terms and nonlinear robin boundary conditions the proof of the result is made by combining galerkin method with a priori estimates for this kind of problem

optimality of multilevel preconditioners for local mesh refinement in three dimensions in this article we establish optimality of the bramble pasciak xu bpx norm equivalence and optimality of the wavelet modified or stabilized hierarchical basis whb preconditioner in the setting of local 3d mesh refinement in the analysis of whb methods a critical first step is to establish the optimality of bpx norm equivalence for the refinement procedures under consideration while the available optimality results for the bpx norm have been constructed primarily in the setting of uniformly refined meshes a notable exception is the local 2d red green result due to dahmen and kunoth the purpose of this article is to extend this original 2d optimality result to the local 3d red green refinement procedure introduced by bornemann erdmann and kornhuber and then to use this result to extend the whb optimality results from the quasi uniform setting to local 2d and 3d red green refinement scenarios the bpx extension is reduced to establishing that locally enriched finite element subspaces allow for the construction of a scaled basis which is formally riesz stable this construction turns out to rest not only on the shape regularity of the refined elements but also critically on a number of geometrical properties we establish between neighboring simplices produced by the bornemann erdmann kornhuber bek refinement procedure it is possible to show that the number of degrees of freedom used for smoothing is bounded by a constant times the number of degrees of freedom introduced at that level of refinement indicating that a practical implementable version of the resulting bpx preconditioner for the bek refinement setting has provably optimal linear computational complexity per iteration an interesting implication of the optimality of the whb preconditioner is the a priori h1 stability of the l2 projection the existing a posteriori approaches in the literature dictate a reconstruction of the mesh if such conditions cannot be satisfied the theoretical framework employed supports arbitrary spatial dimension d geq 1 and requires no coefficient smoothness assumptions beyond those required for well posedness in h1

on the singular second order elliptic equation abstract let m g be a smooth and compact riemannian manifold with dimensions n 3 this paper deals with the existence of nontrivial solutions for a nonlinear second order elliptic equation with a singular term and critical sobolev growth we provide a geometric application of this singular equation

on the evolution and qualitative behaviors of bifurcation curves for a boundary value problem abstract we study the evolution and qualitative behaviors of bifurcation curves of positive solutions for u x u 1 sin u u p 1 x 1 u 1 u 1 0 where 0 is a bifurcation parameter and p 1 is an evolution parameter on the u plane we prove that the bifurcation curve has exactly one turning point where the curve turns to the left for p 2 it is a monotone curve for p 2 it has at least two turning points for 1 p p 1 2 cos 1 sin 1 1 2 cos 1 sin 1 1 629 and it has infinitely many turning points for p 1 hence we are able to determine the exact number of positive solutions in particular we give complete descriptions of the structure of bifurcation curves when p 2 we also give some numerical simulations of bifurcation curves for p 1

the nehari manifold for a semilinear elliptic problem with the nonlinear boundary condition abstract using the nehari manifold and fibering maps we prove the existence theorem of the nonlinear boundary problem u u u p 2 u for x u n u q 2 u for x on a bounded domain r n we discuss how the nehari manifold changes as changes and show how the existence of solutions depends on the properties of the nehari manifold

multilevel preconditioning this paper is concerned with multilevel techniques for preconditioning linear systems arising from galerkin methods for elliptic boundary value problems a general estimate is derived which is based on the characterization of besov spaces in terms of weighted sequence norms related to corresponding multilevel expansions the result brings out clearly how the various ingredients of a typical multilevel setting affect the growth rate of the condition numbers in particular our analysis indicates how to realize even uniformly bounded condition numbers for example the general results are used to show that the bramble pasciak xu preconditioner for piecewise linear finite elements gives rise to uniformly bounded condition numbers even when the refinements of the underlying triangulations are highly nonuniform furthermore they are applied to a general multivariate setting of refinable shift invariant spaces in particular covering those induced by various types of wavelets

existence and multiplicity results for quasilinear elliptic exterior problems with nonlinear boundary conditions abstract in this paper we study a class of quasilinear elliptic exterior problems with nonlinear boundary conditions existence of ground states and multiplicity results are obtained via variational methods

existence of infinitely many spike solutions for a critical henon type biharmonic equation abstract following the idea of wei and yan 32 with new ingredients to take care of the critical dimension n 5 we construct infinitely many solutions of the biharmonic equation 2 v x v n 4 n 4 in the unit ball of r n n 5 0 with the navier conditions v v 0 on the boundary

on a nonlinear fourth order elliptic equation involving the critical sobolev exponent in this paper we use an algebraic topological argument due to bahri and coron to show how the topology of the domain influences the existence of positive solutions of the following problem involving the bilaplacian operator with the critical sobolev exponent 2u un 4 n 4 in u 0 in u u 0 on where is a bounded domain of rn n 5 with a smooth boundary

positive solutions of semilinear biharmonic equations with critical sobolev exponents in this paper we study the critical growth biharmonic problem with a parameter and establish uniform lower bounds for which is the supremum of the set of related to the existence of positive solutions of the biharmonic problem

bpx preconditioners for the bidomain model of electrocardiology the aim of this work is to develop a bpx preconditioner for the bidomain model of electrocardiology this model describes the bioelectrical activity of the cardiac tissue and consists of a system of a non linear parabolic reaction diffusion partial differential equation pde and an elliptic linear pde modeling at macroscopic level the evolution of the transmembrane and extracellular electric potentials of the anisotropic cardiac tissue the evolution equation is coupled through the non linear reaction term with a stiff system of ordinary differential equations the so called membrane model describing the ionic currents through the cellular membrane the discretization of the coupled system by finite elements in space and semi implicit finite differences in time yields at each time step the solution of an ill conditioned linear system the goal of the present study is to construct analyze and numerically test a bpx preconditioner for the linear system arising from the discretization of the bidomain model optimal convergence rate estimates are established and verified by two and three dimensional numerical tests on both structured and unstructured meshes moreover in a full heartbeat simulation on a three dimensional wedge of ventricular tissue the bpx preconditioner is about 35 faster in terms of cpu times than ilu 0 and an algebraic multigrid preconditioner

balancing neumann neumann methods for the cardiac bidomain model balancing neumann neumann preconditioners are constructed analyzed and numerically studied for the cardiac bidomain model in three dimensions this reaction diffusion system is discretized by low order finite elements in space and implicit explicit methods in time yielding very ill conditioned linear systems that must be solved at each time step the proposed algorithm is based on decomposing the domain into nonoverlapping subdomains and on solving iteratively the bidomain schur complement obtained by implicitly eliminating the degrees of freedom interior to each subdomain the iteration is preconditioned by a balancing neumann neumann method employing local neumann solves on each subdomain and a coarse bidomain solve a novel approach for the estimation of the average operator of the nonoverlapping decomposition provides a framework for designing coarse spaces for balancing neumann neumann methods the theoretical estimates obtained show that the proposed method is scalable quasi optimal and robust with respect to possible coefficient discontinuities of the bidomain operator the results of extensive parallel numerical tests in three dimensions confirm the convergence rates predicted by the theory

damage detection in elastic structures using vibratory residual forces and weighted sensitivity a methodology is presented for detecting structural damage in elastic structures by nondestructiv e means measured modal test data along with a correlated analytical structural model are used to locate potentially damaged regions using residual modal force vectors and to conduct a weighted sensitivity analysis to assess the extent of mass and or stiffness variations where damage is characterized as a stiffness reduction the current approach is unique among other approaches in that it accounts for 1 variations in system mass system stiffness and mass center locations 2 perturbations of both the natural frequencies and modal vectors and 3 statistical confidence factors for the structural parameters and potential experimental instrumentation error moreover this procedure can be used with either full or reduced models a wide variety of numerical examples are presented that show that the current method provides a precise indication of both the location and the extent of structural damage lexible space structures launch vehicles and satellites are susceptible to structural damage over their operating lives from impact operating loads and fatigue undetected and uncorrected damage can lead to structural deterioration of their members and consequently jeopardize the flight safety of the structure thus numerous damage inspection methods and monitoring procedures have been developed examples include x ray ultrasonic testing magnetic resonance coin tap dye penetration and visual inspection these methods can be time consuming and are local assessments often requir ing the exposure of structural elements to the inspector and equipment for detecting damage as a monitoring system for detecting damage of spacecraft in orbit i e space station freedom none of these methods are appropriate an alterna tive approach which formed the basis for our current study is to recognize the fact that modal vibration test data struc tural natural frequencies and mode shapes characterize the state of the structure 1 therefore postflight and in flight e g monitoring data can be used to distinguish whether changes damage have occurred to the structure by comparing these data with a set of baseline data modal testing as a means of inspection has several advantages direct exposure of struc tural elements is not required and at the same time more of the complete structure can be inspected in one modal test by having appropriately placed sensors the consequences of this are a reduction in schedule and cost the use of vibration test data to locate structural damage has been attempted to evaluate the integrity of offshore oil and gas platforms 2 3 composite laminates 4 continuum struc tures 5 8 and recently large space structures 9 11 the early ana lytical and experimental approaches2 3 had limited success as a result of inadequate instrumentation and addressing only fre quency shifts associated with stiffness changes while ignoring changes in the modal vectors the continuum based meth ods5 8 provide valuable insight into the magnitude of fre quency shifts but their usefulness is severely limited to simple structures bars beams plates with unrealistic boundary con

a complete classification of bifurcation diagrams of classes of multiparameter p laplacian boundary value problems we study the bifurcation diagrams of positive solutions of the multiparameter p laplacian problem p u x f u x 0 1 x 1 u 1 u 1 0 where p 1 p y y p 2y p u is the one dimensional p laplacian f u g u h u and 0 and 0 are two bifurcation parameters 0 and 0 are two given real numbers assuming that functions g and h satisfy hypotheses h1 h3 and h4a resp h1 h3 and h4b for fixed 0 resp 0 we give a classification of totally eight qualitatively different bifurcation diagrams we prove that on the u plane resp u plane each bifurcation diagram consists of exactly one curve which is either a monotone curve or has exactly one turning point where the curve turns to the right hence the problem has at most two positive solutions for each 0 resp 0 more precisely we prove the exact multiplicity of positive solutions in addition for all p 1 we give interesting examples which show complete evolution of bifurcation diagrams as resp varies

bpx preconditioning for isogeometric analysis we consider elliptic pdes partial differential equations in the framework of isogeometric analysis i e we treat the physical domain by means of a b spline or nurbs mapping which we assume to be regular the numerical solution of the pde is computed by means of tensor product b splines mapped onto the physical domain we construct additive multilevel preconditioners and show that they are asymptotically optimal i e the spectral condition number of the resulting stiffness matrix is independent of h together with a nested iteration scheme this enables an iterative solution scheme of optimal linear complexity the theoretical results are substantiated by numerical examples in two and three space dimensions

structural damage detection of space truss structures using best achievable eigenvectors a method is presented by which measured modes and frequencies from a modal test can be used to determine the location and magnitude of damage in a space truss structure the damage is located by computing the euclid ean distances between the measured mode shapes and the best achievable eigenvectors the best achievable eigenvectors are the projection of the measured mode shapes onto the subspace defined by the refined analytical model of the structure and the measured frequencies loss of both stiffness and mass properties can be located and quantified to examine the performance of the method when experimentally measured modes are employed various damage detection studies using a laboratory eight bay truss structure were conducted the method per forms well even though the measurement errors inevitably make the damage location more difficult uture structures in space will be orders of magnitude larger and more complex than their predecessors structures such as space station freedom will typically be built around a large flexi ble frame and consist of habitation and experiment modules flexi ble and articulating appendages and numerous utility trays and pay loads along with the need to design the spacecraft to be light weight strong and modular for ease of expansion repair and modification the complexity and size of these structures require that the integrity of the structures be monitored periodically several researchers have studied the possibility of detecting damage or structural anomalies to large space trusses and locating the site of this damage based on changes in the natural frequencies and mode shapes of the structures references 1 and 2 provide a comprehensive overview of many of the proposed methods in this respect and the issues involved with on orbit modal identification and structural verification there are several distinct classes of model improvement damage location methods each of these methods begins with an initial finite element representation of the structure in the case of model improvement the goal is to use test data of the actual structure to modify this initial finite element model so that it may serve as an accurate representation of the un damaged structure in the case of damage location the goal is to use test data from the damaged structure and the updated corre lated finite element model of the undamaged structure to deter mine changes to the stiffness and or mass matrices that would ac count for the observed differences in the measured modes and frequencies the class of update methods that has been explored most widely is known as optimal matrix modification the goal of the optimal matrix modification is to find an updated matrix stiffness and or mass that is as close as possible to the original matrix assumed to be almost correct that accurately produces the measured modes

compactness along the branch of semistable and unstable solutions for an elliptic problem with a singular nonlinearity we study the branch of semistable and unstable solutions i e those whose morse index is at most 1 of the dirichlet boundary value problem u f x 1 u 2 on a bounded domain n which models among other things a simple electrostatic microelectromechanical system mems device we extend the results of 11 relating to the minimal branch by obtaining compactness along unstable branches for 1 n 7 on any domain and for a large class of permittivity profiles f we also show the remarkable fact that powerlike profiles f x x can push back the critical dimension n 7 of this problem by establishing compactness for the semistable branch on the unit ball also for n 8 and as long as as a byproduct we are able to follow the second branch of the bifurcation diagram and prove the existence of a second solution for in a natural range in all these results the conditions on the space dimension and on the power of the profile are essentially sharp 2007 wiley periodicals inc

three positive solutions for dirichlet problems involving critical sobolev exponent and sign changing weight abstract in this paper we study the decomposition of the nehari manifold via the combination of concave and convex nonlinearities furthermore we use this result and the lusternik schnirelman category to prove that a semilinear elliptic equation involving a sign changing weight function has at least three positive solutions

nonlinear elliptic equations with an asymptotically linear reaction term abstract we examine nonlinear elliptic problems driven by the p laplacian and with a reaction term which asymptotically at exhibits a p 1 linear growth 1 p using tools from critical point theory and from morse theory we prove existence theorems and we provide information about the sign of the solutions