the geometry of r adaptive meshes generated using optimal transport methods the principles of mesh equidistribution and alignment play a fundamental role in the design of adaptive methods and a metric tensor and mesh metric are useful theoretical tools for understanding a method s level of mesh alignment or anisotropy we consider a mesh redistribution method based on the monge ampere equation which combines equidistribution of a given scalar density function with optimal transport it does not involve explicit use of a metric tensor although such a tensor must exist for the method and an interesting question to ask is whether or not the alignment produced by the metric gives an anisotropic mesh for model problems with a linear feature and with a radially symmetric feature we derive the exact form of the metric which involves expressions for its eigenvalues and eigenvectors the eigenvectors are shown to be orthogonal and tangential to the feature and the ratio of the eigenvalues corresponding to the level of anisotropy is shown to depend both locally and globally on the value of the density function and the amount of curvature we thereby demonstrate how the optimal transport method produces an anisotropic mesh along a given feature while equidistributing a suitably chosen scalar density function numerical results are given to verify these results and to demonstrate how the analysis is useful for problems involving more complex features including for a non trivial time dependant nonlinear pde which evolves narrow and curved reaction fronts

variable node elements for non matching meshes by means of mls moving least square scheme a new class of finite elements is described for dealing with non matching meshes for which the existing finite elements are hardly efficient the approach is to employ the moving least square mls scheme to devise a class of elements with an arbitrary number of nodal points on the parental domain this approach generally leads to elements with the rational shape functions which significantly extends the function space of the conventional finite element method with a special choice of the nodal points and the base functions the method results in useful elements with the polynomial shape functions for which the c 1 continuity breaks down across the boundaries between the subdomains comprising one element the present scheme possesses an extremely high potential for applications which deal with various problems with discontinuities such as material inhomogeneity crack propagation phase transition and contact mechanics the effectiveness of the new elements for handling the discontinuities due to non matching interfaces is demonstrated using appropriate examples

a multiscale projection method for macro microcrack simulations we present a new multiscale method for crack simulations this approach is based on a two scale decomposition of the displacements and a projection to the coarse scale by using coarse scale test functions the extended finite element method xfem is used to take into account macrocracks as well as microcracks accurately the transition of the field variables between the different scales and the role of the microfield in the coarse scale formulation are emphasized the method is designed so that the fine scale computation can be done independently of the coarse scale computation which is very efficient and ideal for parallelization several examples involving microcracks and macrocracks are given it is shown that the effect of crack shielding and amplification for crack growth analyses can be captured efficiently copyright 2007 john wiley sons ltd

moving mesh methods for problems with blow up in this paper we consider the numerical solution of pdes with blow up for which scaling invariance plays a natural role in describing the underlying solution structures it is a challenging numerical problem to capture the qualitative behaviour in the blow up region and the use of nonuniform meshes is essential we consider moving mesh methods for which the mesh is determined using so called moving mesh partial differential equations mmpdes specifically the underlying pde and the mmpde are solved for the blow up solution and the computational mesh simultaneously motivated by the desire for the mmpde to preserve the scaling invariance of the underlying problem we study the effect of different choices of mmpdes and monitor functions it is shown that for suitable ones the mmpde solution evolves towards a moving mesh which close to the blow up point automatically places the mesh points in such a manner that the ignition kernel which is well known to be a natural coordinate in describing the behaviour of blow up approaches a constant as t to t the blow up time several numerical examples are given to verify the theory for these mmpde methods and to illustrate their efficacy

mls based variable node elements compatible with quadratic interpolation part i formulation and application for non matching meshes two dimensional variable node elements compatible with quadratic interpolation are developed using the moving least square mls approximation the mapping from the parental domain to the physical element domain is implicitly obtained from mls approximation with the shape functions and their derivatives calculated and saved only at the numerical integration points it is shown that the present mls based variable node elements meet the patch test if a sufficiently large number of integration points are employed for numerical integration the cantilever problem with non matching meshes is chosen to check the feasibility of the present mls based variable node elements and the result is compared with that from the lower order case compatible with linear interpolation copyright 2005 john wiley sons ltd

optimal mass transport for higher dimensional adaptive grid generation in this work we describe an approach for higher dimensional adaptive grid generation based on solving the l 2 monge kantorovich problem mkp which is a special case of the classical optimal mass transportation problem two methods are developed for computing the coordinate transformation used to define the grid adaptation for the first method the transformation is determined by solving a parabolic monge ampere equation for a steady state solution for the second method the grid movement is determined from the velocity field obtained by solving a fluid dynamics formulation of the l 2 mkp several numerical experiments are presented to demonstrate the performance of the mkp methods and to compare them with some related adaptive grid methods the experimental results demonstrate that the mkp methods show promise as effective and reliable methods for higher dimensional adaptive grid generation

conservative load transfer along curved fluid solid interface with non matching meshes we investigate the effect of curvature on the accuracy of schemes used to transfer loads along the interface in coupled fluid solid simulations involving non matching meshes we analyze two types of load transfer schemes for the coupled system a point to element projection schemes and b common refinement schemes the accuracy of these schemes over the curved interface is assessed with the aid of static and transient problems we show that the point to element projection schemes may yield inaccurate load transfer from the source fluid mesh to the target solid mesh leading to a weak instability in the form of spurious oscillations and overshoots in the interface solution the common refinement scheme resolves this problem by providing an accurate transfer of discrete interface conditions across non matching meshes we show theoretically that the accurate transfer preserves the stability of the coupled system while maintaining the energy conservation over a reference interface finally we introduce simple analytical error functions which correlate well with the numerical errors of the load transfer schemes

parabolic monge ampere methods for blow up problems in several spatial dimensions this paper constructs and analyses an adaptive moving mesh scheme for the numerical simulation of singular pdes in one or more spatial dimensions the scheme is based on computing a legendre transformation from a regular to a spatially non uniform mesh via the solution of a relaxed form of the monge ampere equation the method is shown to preserve the inherent scaling properties of the pde and to identify natural computational coordinates numerical examples are presented in one and two dimensions which demonstrate the effectiveness of this approach

universality and complexity in cellular automata abstract cellular automata are discrete dynamical systems with simple construction but complex self organizing behaviour evidence is presented that all one dimensional cellular automata fall into four distinct universality classes characterizations of the structures generated in these classes are discussed three classes exhibit behaviour analogous to limit points limit cycles and chaotic attractors the fourth class is probably capable of universal computation so that properties of its infinite time behaviour are undecidable

development of three dimensional interface elements for coupling of non matching hexahedral meshes gluing of a non matching interface between dissimilar meshes is one of challenging tasks in computational mechanics the interface element method iem has been developed using an appropriate interpolation scheme between partitioned finite element meshes the continuity of displacements across the interface is satisfied exactly and the completeness of interface element shape functions provides a reasonable transfer of strain fields between dissimilar finite element meshes research related to the iem has been mainly accomplished in two dimensions because the construction of interface elements in three dimensions is not as simple as the case of two dimensions in this paper a new scheme is presented to construct the interface elements for connecting dissimilar hexahedral finite element meshes with an arbitrarily curved non matching interface element edges crossing any other edges in an interface region are modified into continuous intersection polygons to eliminate overlays and gaps between dissimilar meshes and these polygons are divided into triangles to create prism shaped interface elements numerical experiments on a linear elasticity problem with non matching meshes illustrate the effectiveness of the method

moving mesh generation using the parabolic monge ampere equation this article considers a new method for generating a moving mesh which is suitable for the numerical solution of partial differential equations pdes in several spatial dimensions the mesh is obtained by taking the gradient of a scalar mesh potential function which satisfies an appropriate nonlinear parabolic partial differential equation this method gives a new technique for performing r adaptivity based on ideas from optimal transportation combined with the equidistribution principle applied to a time varying scalar monitor function used successfully in moving mesh methods in one dimension detailed analysis of this new method is presented in which the convergence regularity and stability of the mesh is studied additionally this new method is shown to be straightforward to program and implement requiring the solution of only one simple scalar time dependent equation in arbitrary dimension with adaptivity along the boundaries handled automatically we discuss three preexisting methods in the context of this work examples are presented in which either the monitor function is prescribed in advance or it is given by the solution of a partial differential equation

mls based variable node elements compatible with quadratic interpolation part ii application for finite crack element two dimensional finite crack elements for simulation of propagating cracks are developed using the moving least square mls approximation the mapping from the parental domain to the physical element domain is implicitly obtained from mls approximation with the shape functions and their derivatives calculated and saved only at the numerical integration points the mls based variable node elements are extended to construct the crack elements which allow the discontinuity of crack faces and the crack tip singularity the accuracy of the crack elements is checked by calculating the stress intensity factor under mode i loading the crack elements turn out to be very efficient and accurate for simulating crack propagations only with the minimal amount of element adjustment and node addition as the crack tip moves numerical results and comparison to the results from other works demonstrate the effectiveness and accuracy of the present scheme for the crack elements copyright 2005 john wiley sons ltd

moving mesh methods based on moving mesh partial differential equations several versions of a moving mesh method are developed based on a mesh spatial smoothing technique and on the moving mesh pdes derived in a previous paper these versions are quite simple and easy to program they are applied to three bench mark one dimensional problems which show different solution behaviour the numerical results clearly demonstrate that the present methods are capable of accurately tracking rapid spatial and temporal transitions

a new three dimensional variable node finite element and its application for fluid solid interaction problems abstract a new three dimensional variable node element is proposed to achieve an efficient connection of non matching interfaces between dissimilar finite element meshes the variable node elements at interfaces are compatible with conventional finite elements and satisfy the interface conditions of continuity and completeness across non matching meshes the present method thereby provides a seamless connection between dissimilar finite element meshes and a correct load transfer across non matching interfaces therefore it leads to an efficient approach for solving a class of problems involving the non matching interfaces such as fluid solid interaction problems numerical examples are presented to show the effectiveness of the present method for solving fluid solid interaction problems with non conforming interfaces and the usefulness and application of the present method are discussed

an efficient numerical method for the solution of the l_2 optimal mass transfer problem in this paper we present a new computationally efficient numerical scheme for the minimizing flow approach for the computation of the optimal l 2 mass transport mapping in contrast to the integration of a time dependent partial differential equation proposed in s angenent s haker and a tannenbaum siam j math anal 35 2003 pp 61 97 we employ in the present work a direct variational method the efficacy of the approach is demonstrated on both real and synthetic data

a new coupling strategy for fluid solid interaction problems by using the interface element method this paper presents a new approach to simulate fluid solid interaction fsi problems involving non matching meshes the coupling between the fluid and solid domains with dissimilar finite element meshes consisting of 4 node quadrilateral elements is achieved by using the interface element method continuity and compatibility conditions across the interface between fluid and solid meshes are satisfied exactly by introducing the interface elements defined on an interfacing region importantly a consistent transfer of loads through the interface elements guarantees the present method to be an efficient approach of the solution strategy to fsi problems an arbitrary lagrangian eulerian description is adopted for the fluid domain while for the solid domain an updated lagrangian formulation is considered to accommodate finite deformations of an elastic structure the fluid equations for velocity and pressure and the solid equations for velocity are strongly coupled in a single computational domain by means of the interface elements numerical results are presented for fsi problems involving non matching meshes to demonstrate the effectiveness of the methodology copyright 2009 john wiley sons ltd

analysis of moving mesh partial differential equations with spatial smoothing two moving mesh partial differential equations mmpdes with spatial smoothing are derived based upon the equidistribution principle this smoothing technique is motivated by the robust moving mesh method of dorfi and drury j comput phys 69 1987 pp 175 195 it is shown that under weak conditions the basic property of no node crossing is preserved by the spatial smoothing and a local quasi uniformity property of the coordinate transformations determined by these mmpdes is proven it is also shown that discretizing the mmpdes using centered finite differences these basic properties are preserved

review of coupling methods for non matching meshes abstract domain decomposition is nowadays a common way to speed up complex computations however the discrete meshes used in the different domains do not have to match at their common interface especially when different physical fields are involved such as in fluid structure interaction computations exchange of information over this interface is therefore no longer trivial in this paper six methods that can deal with the information transfer between non matching meshes in fluid structure interaction computations are compared for different criteria this is done for analytical test cases as well as a quasi 1d fluid structure interaction problem two methods based on radial basis functions one with compact support and one using thin plate splines are favoured over the other methods because of their high accuracy and efficiency

on the time continuous mass transport problem and its approximation by augmented lagrangian techniques in j d benamou and y brenier numer math 84 2000 pp 375 393 a computational fluid dynamic approach was introduced for computing the optimal map occurring in the monge kantorovich problem though the described augmented lagrangian method involves a hilbertian framework the discussion was purely formal taking advantage of the recent progress in optimal transport theory l a caffarelli comm pure appl math 45 1992 pp 1141 1151 l a caffarelli ann of math 2 144 1996 pp 453 496 d cordero erausquin c r acad sci paris ser i math 329 1999 pp 199 202 r j mccann geom funct anal 11 2001 pp 589 608 and despite the lack of coercivity of the hilbertian problem we establish an existence result then under a reasonable assumption of positivity for the density we prove the existence of saddle points for both lagrangians defined in benamou and brenier and finally prove the convergence of the numerical method

global properties of cellular automata cellular automata are discrete mathematical systems that generate diverse often complicated behavior using simple deterministic rules analysis of the local structure of these rules makes possible a description of the global properties of the associated automata a class of cellular automata that generate infinitely many aperiodic temporal sequences is defined as is the set of rules for which inverses exist necessary and sufficient conditions are derived characterizing the classes of nearest neighbor rules for which arbitrary finite initial conditions i evolve to a homogeneous state ii generate at least one constant temporal sequence

moving mesh strategy based on a gradient flow equation for two dimensional problems in this paper we introduce a moving mesh method for solving pdes in two dimensions it can be viewed as a higher dimensional generalization of the moving mesh pde mmpde strategy developed in our previous work for one dimensional problems w huang y ren and r d russell siam j numer anal 31 1994 pp 709 730 the mmpde is derived from a gradient flow equation which arises using a mesh adaptation functional in turn motivated from the theory of harmonic maps geometrical interpretations are given for the gradient equation and functional and basic properties of this mmpde are discussed numerical examples are presented where the method is used both for mesh generation and for solving time dependent pdes the results demonstrate the potential of the mesh movement strategy to concentrate the mesh points so as to adapt to special problem features and to also preserve a suitable level of mesh orthogonality

partitioned formulation of internal fluid structure interaction problems by localized lagrange multipliers abstract a partitioned formulation of compressible internal fluid structure interaction problems is presented by employing a displacement model for both the fluid and structure partitioning is effected by a localized version of the method of lagrange multipliers which assigns two independent sets of lagrange multipliers to the structural and fluid interfaces two major features of the present formulation include an interface compliance normalization that helps capture the predominant physics of interaction phenomena when the interfaces are characterized by two radically different rigidities and a novel transformation of the displacement model into a fluid pressure model that is suitable for both transient and vibration analyses the present formulation first solves for the interface lagrange multipliers which are subsequently used to solve for the structural displacements and the fluid displacement or pressure by employing two independent analysis modules

immersed finite element method abstract in this paper the immersed finite element method ifem is proposed for the solution of complex fluid and deformable structure interaction problems encountered in many physical models in ifem a lagrangian solid mesh moves on top of a background eulerian fluid mesh which spans over the entire computational domain hence the mesh generation is greatly simplified moreover both fluid and solid domains are modeled with the finite element methods and the continuity between the fluid and solid sub domains are enforced via the interpolation of the velocities and the distribution of the forces with the reproducing kernel particle method rkpm delta function in comparison with the immersed boundary ib method the higher ordered rkpm delta function enables the fluid domain to have nonuniform spatial meshes with arbitrary geometries and boundary conditions the use of such kernel functions may eventually open doors to multi scale and multi resolution modelings of complex fluid structure interaction problems rigid and deformable spheres dropping in channels are simulated to demonstrate the unique capabilities of the proposed method the results compare well with the experimental data to the authors knowledge these are the first solutions that deal with particulate flows with very flexible solids

a new computational approach to contact mechanics using variable node finite elements in this paper a new computational strategy for two dimensional contact problems is developed with the aid of variable node finite elements within the range of infinitesimal deformations the variable node elements which are among mls moving least square based finite elements enable us to transform node to surface contact problems into node to node contact problems this contact formulation with variable node elements leads to an accurate and effective solution procedure needless to mention that the contact patch test is passed without any additional treatment through several numerical examples we demonstrate its simplicity and the effectiveness of the proposed scheme

the algebraic properties of finite cellular automata abstract the basic patterns by cellular automata have been phenomenologically classified into four types by wolfram i propose a new algebraic method to predict the types of patterns by the elementary cellular automata this method is one example of the application of brownian algebra which is very important in the area of cognitive science the classification carried out by this method is in good agreement with wolfram s classification a significant difference in the present classification is that class 3 by wolfram is divided into two types this difference is apparent in the property of the filter automata which are constructed by the same rule

meshfree finite difference approximations for functions of the eigenvalues of the hessian we introduce meshfree finite difference methods for approximating nonlinear elliptic operators that depend on second directional derivatives or the eigenvalues of the hessian approximations are defined on unstructured point clouds which allows for very complicated domains and a non uniform distribution of discretisation points the schemes are monotone which ensures that they converge to the viscosity solution of the underlying pde as long as the equation has a comparison principle numerical experiments demonstrate convergence for a variety of equations including problems posed on random point clouds complex domains degenerate equations and singular solutions

a new dual mortar method for curved interfaces 2d elasticity dual mortar method formulations have shown to be a very effective and efficient way for interfacing e g tying contacting dissimilar meshes on the other hand we have recently found that they can sometimes perform quite poorly when applied to curved surfaces in some solid mechanics applications a new modified two dimensional dual mortar method for piecewise linear finite elements is developed that overcomes this deficiency and is demonstrated on a model problem furthermore mathematical analysis is provided to demonstrate the optimal convergence and stability of the new method copyright 2005 john wiley sons ltd

partitioned procedures for the transient solution of coupled aroelastic problems part i model problem theory and two dimensional application abstract in order to predict the dynamic response of a flexible structure in a fluid flow the equations of motion of the structure and the fluid must be solved simultaneously in this paper we present several partitioned procedures for time integrating this focus coupled problem and discuss their merits in terms of accuracy stability heterogeneous computing i o transfers subcycling and parallel processing all theoretical results are derived for a one dimensional piston model problem with a compressible flow because the complete three dimensional aeroelastic problem is difficult to analyze mathematically however the insight gained from the analysis of the coupled piston problem and the conclusions drawn from its numerical investigation are confirmed with the numerical simulation of the two dimensional transient aeroelastic response of a flexible panel in a transonic non linear euler flow regime

the monge ampere trajectory correction for semi lagrangian schemes requiring that numerical estimates of the flow trajectories comply with the fundamental euler expansion formula that governs the evolution of a volume of fluid leads to a second order nonlinear monge ampere partial differential equation mae in cossette and smolarkiewicz 2011 15 a numerical algorithm based on solving the mae with an inexact newton krylov solver has been developed and used to correct standard estimates of the departure points of flow trajectories provided by a classical semi lagrangian scheme in the context of an incompressible fluid here we extend the theoretical analysis of the elemental rotational and deformational motions presented in cossette and smolarkiewicz 2011 15 in particular closed form analytic solutions are derived for both cases that serve to illustrate the mechanics of the enhanced trajectory scheme and to address the issues of existence and uniqueness scalar advection shows that the ma correction improves mass conservation substantially by suppressing anomalous fluid contraction the impact of the ma correction on complex flows is studied in the framework of an ideal magneto fluid in which the formation of current sheets leads to topological changes and reconnection of field lines the use of the ma correction improves the numerical stability of the solutions and prevents trajectory intersections as well as the spurious growth of the magnetic energy 2d and 3d examples are presented and the computational performance of the solver is documented

an improved interface element with variable nodes for non matching finite element meshes extensive improvements of the interface element method iem are proposed for the efficient treatment of non matching finite element meshes our approach enables us to establish the master element via the moving least square mls approximation and so to remove the cumbersome process of constructing interface elements the values of shape functions and their derivatives are therefore mapped from the master element as in the conventional finite element method for the assurance of convergence and compatibility condition a patch test is demonstrated through several examples of 2d linear elasticity the convergence rate is compared between the present interface element and the previous version

moving mesh partial differential equations mmpdes based on the equidistribution principle this paper considers several moving mesh partial differential equations that are related to the equidistribution principle several of these are new and some correspond to discrete moving mesh equations that have been used by others their stability is analyzed and it is seen that a key term for most of these moving mesh pdes is a source like term that measures the level of equidistribution it is shown that under weak assumptions mesh crossing cannot occur for most of them finally numerical experiments for these various moving mesh pdes are performed to study their relative properties

measuring mesh qualities and application to variational mesh adaptation the mesh assessment problem is investigated in this paper by taking into account the shape and size of elements and the solution behavior three elementwise mesh quality measures characterizing the shape alignment and adaptation features of elements are introduced according to the estimates of interpolation error developed on a general mesh an adaptive mesh is assessed by an overall quality measure defined as a weighted lebesgue norm of a product of the three elementwise quality measures it is shown that the overall quality of a mesh is good if the overall mesh quality measure is small or significantly smaller than the so called roughness measure of the solution defined as the ratio of two lebesgue norms of a derivative of the solution the definition of the overall mesh quality measure comes in such a way that the measure appears in the underlying error bound as the only factor depending substantially on the mesh as an immediate result the task of mesh adaptation becomes to control the overall mesh quality this idea is applied to variational mesh adaptation to develop two functionals one new and the other related to an existing functional recently developed using the regularity and equidistribution arguments numerical experiments are given to demonstrate the ability of the functionals to generate adaptive meshes of good quality

efficient mesh deformation based on radial basis function interpolation by means of the inverse fast multipole method abstract radial basis function interpolation is often employed in mesh deformation algorithms for unstructured meshes for example in fluid structure interaction or design optimization problems this is known to be a robust methodology that results in high quality deformed meshes the applicability of this method to large problems is currently hampered by its prohibitive computational cost however which is caused by the need to solve a dense system of equations the computation time grows as o n b 3 if a direct solver is employed for the solution where n b denotes the number of boundary nodes in the mesh while alternative iterative solvers often suffer from an unfavorable convergence behavior in this paper we present the inverse fast multipole method as a novel fast approximate direct solver with a computational cost scaling as o n b the linear complexity is achieved by transforming the dense system into an extended sparse system along with the compression of certain matrix blocks into low rank factorizations the solver is inexact although the error can be controlled and made as small as needed a low accuracy solver can hence be used as an efficient preconditioner in an iterative scheme numerical benchmarks are presented demonstrating that the proposed approach enhances the computational efficiency of mesh deformation algorithms based on radial basis function interpolation significantly without jeopardizing their robustness and quality

a positivity preserving adaptive moving mesh method for cancer cell invasion models abstract in this paper we present an efficient adaptive moving mesh finite difference method for the modeling of early stage cancer cell invasion of tissue or extracellular matrix ecm the cancer cell invasion model is a nonlinear system of reaction diffusion taxis partial differential equations pdes describing the time evolution of the cancer cells density and concentrations of proteins of the ecm the solution of the model exhibits very rapid variations at the boundary of the healthy and cancer cells thus using a uniform grid method to solve the cancer cell invasion model requires a very large number of grid points in order to resolve the steep gradients of the solution precisely as a result the computation can become prohibitively expensive and inefficient in this effort we propose a positivity preserving scheme for the spatial discretization of the model on an adaptive mesh to accurately capture the sharp structures of the solution the adaptive mesh is generated by a coordinate transformation obtained as the solution of the optimal mass transfer problem several numerical experiments are presented to demonstrate the performance of the adaptive mesh method for solving the cancer cell invasion model the numerical results show that the proposed adaptive mesh method is effective in capturing and predicting the correct behavior of the cancer cells movement within the ecm

mls moving least square based finite elements for three dimensional nonmatching meshes and adaptive mesh refinement abstract with the aid of moving least square mls approximation a new class of three dimensional finite elements are proposed for treating nonmatching meshes and adaptive mesh refinement for which the existing finite elements are hardly efficient with a special choice of the weight function supports and the base functions the method results in useful elements with the polynomial shape function for which the c 1 continuity breaks down on the boundaries between the neighboring subdomains comprising one element the effectiveness of the new elements in handling the discontinuities due to nonmatching interfaces and automatic mesh refinement is demonstrated via three dimensional examples

an efficient dynamically adaptive mesh for potentially singular solutions abstract we develop an efficient dynamically adaptive mesh generator for time dependent problems in two or more dimensions the mesh generator is motivated by the variational approach and is based on solving a new set of nonlinear elliptic pdes for the mesh map when coupled to a physical problem the mesh map evolves with the underlying solution and maintains high adaptivity as the solution develops complicated structures and even singular behavior the overall mesh strategy is simple to implement avoids interpolation and can be easily incorporated into a broad range of applications the efficacy of the mesh is first demonstrated by two examples of blowing up solutions to the 2 d semilinear heat equation these examples show that the mesh can follow with high adaptivity a finite time singularity process the focus of applications presented here is however the baroclinic generation of vorticity in a strongly layered 2 d boussinesq fluid a challenging problem the moving mesh follows effectively the flow resolving both its global features and the almost singular shear layers developed dynamically the numerical results show the fast collapse to small scales and an exponential vorticity growth

a moving mesh method based on the geometric conservation law a new adaptive mesh movement strategy is presented which unlike many existing moving mesh methods targets the mesh velocities rather than the mesh coordinates the mesh velocities are determined in a least squares framework by using the geometric conservation law specifying a form for the jacobian determinant of the coordinate transformation defining the mesh and employing a curl condition by relating the jacobian to a monitor function one is able to directly control the mesh concentration the geometric conservation law an identity satisfied by any nonsingular coordinate transformation is an important tool which has been used for many years in the engineering community to develop cell volume preserving finite volume schemes it is used here to transform the algebraic expression specifying the jacobian into an equivalent differential relation which is the key formula for the new mesh movement strategy it is shown that the resulting method bears a close relation with the lagrangian method advantages of the new approach include the ease of controlling the cell volumes and therefore mesh adaption and a theoretical guarantee for existence and nonsingularity of the coordinate transformation it is shown that the method may suffer from the mesh skewness a consequence resulting from its close relation with the lagrangian method numerical results are presented to demonstrate various features of the new method

optimal mass transport based adaptive mesh method for phase field models of two phase fluid flows in this paper we present an efficient adaptive mesh method for solving phase field model of a mixture of two phase incompressible fluid flows the adaptive mesh is generated by a coordinate transformation that is determined as the solution of the classical problem of the optimal mass transportation also known as mongekantorovich problem mkp the numerical solution of the mkp is computed by taking the gradient of the steady state solution of a parabolic mongeampre equation pmae several numerical experiments are conducted to demonstrate the performance of the pmae method for solving phase field models the numerical results illustrate that the adaptive mesh computed with pmae method captures well the moving interface of the phase field function

a perturbed lagrangian formulation for the finite element solution of contact problems making use of a perturbed lagrangian formulation a finite element procedure for contact problems is developed for the general case in which node to node contact no longer holds the proposed procedure leads naturally to a discretization of the contact interface into contact segments within the context of a bilinear interpolation for the displacement field a mixed finite element approximation is introduced by assuming discont naous contact pressure constant on the contact segment because of this piece wise constant approximation the gap function enters into the formulation in an average sense instead of through a point wise definition numerical examples are presented that illustrate the performance of the proposed procedure current finite element formulations for contact problems based on either the classical lagrange parameter procedure l 3 12 203 or the penalty function method 4 6 11 are characterized by a point wise enforcement of the contact constraint condition in the sense that penetration of the bodies is established on a nodal basis moreover in this methods the recovery of the contact pressure over the element from the contact nodal forces generally requires an additional procedure within the framework of classical lagrange multiplier methods the contact condition is exactly satisfied by transforming the constrained problem into an unconstrained one with the introduction of additional variables lagrange multipliers these extra variables add computational effort to the solution process which often requires special procedures to handle the presence of zero diagonal terms penalty methods on the other hand enable one to transform the constrained problem into an unconstrained one without introducing additional variables the constraint condition is now satisfied only approximately for finite values of the penalty parameter the main difhculty associated with these methods however lies in the poor conditioning of the problem as the penalty is increased to more accurately enforce the constraint condition this is a well understood phenomenon particularly in the context of the incompressible and nearly incompressible problem in solid and fluid mechanics e g see l 22 25 for a review recently augmented lagrangian procedures have been proposed as a promising way to partially overcome these difficulties and regularize the penalty formulation e g see the survey in 7 8 within the framework of zinearited e u c it is possible to restrict the finite element formulation of contact problems by assuming that node to node contact occurs this is in fact the case often considered in the literature l 2 4 6 12 20 211 in the general context of fully

soliton like behavior in automata abstract we propose a new kind of automaton that uses newly computed site values as soon as they are available we call them filter automata fa they are analogous to infinite impulse response iir digital filters whereas the usual cellular automata ca correspond to finite impulse response fir digital filters it is shown that as a class the fa s are equivalent to ca s in the sense that the same array of space generation values can be produced they must be generated in a different order however a particular class of irreversible totalistic fa s are described that support a profusion of persistent structures that move at different speeds and these particle like patterns collide in nondestructive ways they often pass through one another with nothing more than a phase jump much like the solitons that arise in the solution of certain nonlinear differential equations histograms of speed displacement and period are given for neighborhood radii from 2 to 6 and particles with generators up to 16 bits wide we then present statistics for neighborhood radii 2 to 9 which show that collisions which preserve the identity of particles are very common

consistent mesh tying methods for topologically distinct discretized surfaces in non linear solid mechanics this work presents a new technique for joining dissimilar meshes in non linear finite element analysis with the novel result that the adjoining surfaces do not need to be spatially coincident for patch tests to be passed the work builds in some sense on results previously presented by dohrmann et al international journal for numerical methods in engineering 1991 40 1205 but considerably generalizes upon the results presented there most importantly it is shown that consistency of the tied finite element formulation may be achieved such that implementation of element formulations is not affected through the use of a three field variational principle applied over the contacting region a key conclusion of the work is that although the approach advocated bears some resemblance to mortar element schemes for contact analysis see for example mcdevitt and laursen international journal for numerical methods in engineering 2000 48 1525 a straightforward application of a mortaring scheme is not sufficient to guarantee patch test passage when dissimilar meshing of curved surfaces leads to local open volumes or overlaps between tied surfaces in their reference configurations the proposed algorithm is demonstrated through a number of numerical examples performed within a matrix free non linear equation solving framework published in 2003 by john wiley sons ltd

conservative load projection and tracking for fluid structure problems the loose coupling of computational fluid dynamics and computational structural dynamics solvers introduces some problems related to the information transfer between the codes some techniques developed to solve the problems of the load transfer and interface surface tracking are presented the main criterion is to achieve conservation of total loads and total energy the load projection scheme is based on gaussian integration and fast interpolation algorithms for unstructured grids the surface tracking algorithm also based on interpolation is important for many applications including aeroelastic deformation of wings due to aerodynamic loads the methodologies not only improve present fluid structure interaction simulations but also increase the range of their applicability these techniques are of general character and can be used in other multidisciplinary applications as well

node to node scheme for three dimensional contact mechanics using polyhedral type variable node elements abstract a node to node contact scheme which is applicable for three dimensional contact analyses involving large deformations is developed with the aid of polyhedral elements the key issue is to transform nonmatching meshes into matching meshes in a seamless manner because polyhedral elements are allowed to have arbitrary numbers of polygonal faces and nodes they can be used as transition elements for coupling nonmatching meshes in this paper the polyhedral elements make it possible to always maintain node to node contact during the contact deformation the present approach guarantees that the patch test is passed and the nonpenetration condition is satisfied and hence it yields smoother contact pressure with faster convergence than the conventional node to surface or surface to surface contact scheme

a dual primal feti method for solving a class of fluid structure interaction problems in the frequency domain summary the dual primal finite element tearing and interconnecting method feti dp is extended to systems of linear equations arising from a finite element discretization for a class of fluid structure interaction problems in the frequency domain a preconditioned generalized minimal residual method is used to solve the linear equations for the lagrange multipliers introduced on the subdomain boundaries to enforce continuity of the solution the coupling between the fluid and the structure on the fluid structure interface requires an appropriate choice of coarse level degrees of freedom in the feti dp algorithm to achieve fast convergence several choices are proposed and tested by numerical experiments on three dimensional fluid structure interaction problems in the mid frequency regime that demonstrate the greatly improved performance of the proposed algorithm over the standard feti dp method copyright 2011 john wiley sons ltd

a simple algorithm for localized construction of non matching structural interfaces a simple and effective algorithm for the modular construction of non matched interfaces is presented for the partitioned solution of large scale structural problems the formulation is based on a recently developed four field variational principle which introduces a connection frame between the interfaced partitions a key result of the present study is a frame nodal placement criterion that uniquely determines the frame discretization into piecewise linear elements so that the interface patch test condition is satisfied a priori the method is demonstrated with several 2d and 3d example problems

interpolation revisited medical images application based on the theory of approximation this paper presents a unified analysis of interpolation and resampling techniques an important issue is the choice of adequate basis functions the authors show that contrary to the common belief those that perform best are not interpolating by opposition to traditional interpolation the authors call their use generalized interpolation they involve a prefiltering step when correctly applied the authors explain why the approximation order inherent in any basis function is important to limit interpolation artifacts the decomposition theorem states that any basis function endowed with approximation order ran be expressed as the convolution of a b spline of the same order with another function that has none this motivates the use of splines and spline based functions as a tunable way to keep artifacts in check without any significant cost penalty the authors discuss implementation and performance issues and they provide experimental evidence to support their claims

polar factorization and monotone rearrangement of vector valued functions given a probability space x p and a bounded domain r in rd equipped with the lebesgue measure 1 i normalized so that 10 i i it is shown under additional technical assumptions on x and q that for every vector valued function u e lp x p rd there is a unique polar factorization u v s where is a convex function defined on r and s is a measure preserving mapping from x p into q i i provided that u is nondegenerate in the sense that p u e 0 for each lebesgue negligible subset e of rd through this result the concepts of polar factorization of real matrices helmholtz decomposition of vector fields and nondecreasing rearrangements of real valued functions are unified the monge amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate monge kantorovich problem

constraint relationships in linear and nonlinear finite element analyses finite element analysis of engineering structures commonly requires the use of inter nodal displacement constraints current algorithms for this facility elimination lagrange multipliers and penalty function methods are briefly reviewed an alternative approach is proposed which avoids some of the problems of these methods this technique uses solution of the unconstrained stiffness equation with an extended number of right hand sides the resulting solutions are then combined to satisfy the constraints this method has application in the synthesis of sub structures with incompatible boundary variables it is particularly efficient when the constraint affects many variables but the kinematics of the constraint can be expressed in terms of few independent variables e g rigid body motion of part of the structure in common with the lagrange method it produces data for use in a nonlinear analysis with constraints the formulation of rigid body constraints in geometric nonlinearity is detailed and demonstrated

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100