on the numerical integration of trimmed isogeometric elements abstract a numerical algorithm is proposed to construct efficient quadrature rules for trimmed isogeometric elements of arbitrary shape and topology as part of the standard finite element preprocessing step the constructed integration rule is unique to an element and it is considered to be optimal in the sense that the final quadrature points and weights satisfy the moment fitting equations within the trimmed domain up to a predefined tolerance the resulting quadrature points are in the interior of the trimmed domain and the positivity of the weights is preserved the accuracy and efficiency of the quadrature scheme are assessed and compared to traditionally employed integration techniques results indicate that the proposed integration rules are more precise and inhibit the proliferation of quadrature points observed in competing approaches selected problems of elastostatics elastodynamics and elasto plastic dynamics are used to further demonstrate the validity and efficacy of the method

the degrees of freedom of partial least squares regression the derivation of statistical properties for partial least squares regression can be a challenging task the reason is that the construction of latent components from the predictor variables also depends on the response variable while this typically leads to good performance and interpretable models in practice it makes the statistical analysis more involved in this work we study the intrinsic complexity of partial least squares regression our contribution is an unbiased estimate of its degrees of freedom it is defined as the trace of the first derivative of the fitted values seen as a function of the response we establish two equivalent representations that rely on the close connection of partial least squares to matrix decompositions and krylov subspace techniques we show that the degrees of freedom depend on the collinearity of the predictor variables the lower the collinearity the higher the complexity in particular they are typically higher than the naive approach that defines the degree

appending boundary conditions by lagrange multipliers analysis of the lbb condition this paper is concerned with the analysis of discretization schemes for second order elliptic boundary value problems when essential boundary conditions are enforced with the aid of lagrange multipliers specifically we show how the validity of the ladyskaja babusska brezzi lbb condition for the corresponding saddle point problems depends on the various ingredients of the involved discretizations the main result states that the lbb condition is satisfied whenever the discretization step length on the boundary h gamma sim 2 ell is somewhat bigger than the one on the domain h omega sim 2 j this is quantified through constants stemming from the trace theorem norm equivalences for the multiplier spaces on the boundary and direct and inverse inequalities in order to better understand the interplay of these constants we then specialize the setting to wavelet discretizations in this case the stability criteria can be stated solely in terms of spectral properties of wavelet representations of the trace operator we conclude by illustrating our theoretical findings by some numerical experiments we stress that the results presented here apply to any spatial dimension and to a wide selection of lagrange multiplier spaces which in particular need not be traces of the trial spaces however we do always assume that a hierarchy of nested trial spaces is given

on some techniques for approximating boundary conditions in the finite element method we discuss the stabilization of finite element methods in which essential boundary conditions are approximated by babuska s method of lagrange multipliers and we show that there is a close connection with this technique and a classical method by nitsche

web spline based mesh free finite element analysis for the approximation of the stationary navier stokes problem he objective in this paper is to discuss the existence and the uniqueness of a weighted extended b spline web spline based discrete solution for the stationary incompressible navier stokes equations the web spline discretization is newly developed methodology which satisfies the inf sup condition or ladyshenskaya babus ka brezzi lbb condition the main advantage of these new elements over standard finite elements is that they use regular grids instead of irregular partitions of the domain thus eliminating the difficult and time consuming pre processing step an error estimate for this web spline based discrete solution is also obtained

immersed b spline i spline finite element method for geometrically complex domains a novel b spline based immersed finite element method is introduced for the computation of geometrically and topologically complex problems the geometry description and the finite element analysis rely on a block structured logically cartesian mesh which encloses the domain of interest a signed distance function is used for representing the domain on the cartesian mesh whereby the domain boundary is the zeroth level set of the signed distance function away from the domain boundaries the standard b spline basis functions are used for the finite element interpolation close to domain boundaries a new approach has been developed for modifying the b spline basis functions so that they locally interpolate the dirichlet boundary conditions the efficiency and robustness of the proposed approach is demonstrated with a number of one two and three dimensional linear boundary value problems

stress constrained shape and topology optimization with fixed mesh a b spline finite cell method combined with level set function abstract in this paper we develop an efficient and flexible design method that integrates the b spline finite cell method b spline fcm and the level set function lsf for stress constrained shape and topology optimization any structure of complex geometry is embedded within an extended regular and fixed eulerian mesh no matter how the structure is optimized high order b spline shape functions are further implemented to ensure precisions of stress analysis and sensitivity analysis meanwhile level set functions i e implicit functions are used to enable topological changes of the considered structure through smooth boundary variations involved parameters rather than the conventional discrete form of lsf are directly taken as design variables to facilitate the numerical computing process to be specific the lsf is constructed by means of r functions that incorporate cubic splines as implicit functions to offer flexibilities for shape optimization within the framework of fixed mesh while the compactly supported radial basis functions cs rbfs are employed as implicit functions for stress constrained topology optimization it is shown the proposed fcm lsf method is a convenient approach that makes it possible to calculate stress and stress sensitivities with high precision representative examples of shape and topology optimization with and without stress constraints are solved with success demonstrating the advantages of the fcm lsf method

two stage approximation methods with extended b splines we develop and analyze a framework for two stage methods with eb splines applicable to continuous and discrete approximation problems in particular we propose a weighted discrete least squares fit that yields optimal convergence rates for sufficiently dense data on lipschitz domains in rd

numerical homogenization of the acoustic wave equations with a continuum of scales in this paper we consider numerical homogenization of acoustic wave equations with heterogeneous coefficients namely when the bulk modulus and the density of the medium are only bounded we show that under a cordes type condition the second order derivatives of the solution with respect to harmonic coordinates are l 2 instead h 1 with respect to euclidean coordinates and the solution itself is in l 0 t h 2 instead of l 0 t h 1 with respect to euclidean coordinates then we propose an implicit time stepping method to solve the resulted linear system on coarse spatial scales and present error estimates of the method it follows that by pre computing the associated harmonic coordinates it is possible to numerically homogenize the wave equation without assumptions of scale separation or ergodicity

programming finite element methods with weighted b splines we discuss the main components of the recently developed femb program package which implements finite element methods with weighted b splines for basic linear elliptic boundary value problems in two and three dimensions a three dimensional implementation without topological restrictions has not been available before we describe in particular the mathematical background for the recursive quadrature cubature over boundary cells and explain how to utilize the regular data structure of uniform b splines efficiently considering the lame navier equations of linear elasticity as a typical example we illustrate the performance of the main femb routines the numerical tests confirm that due to the new integration routines the weighted b spline solvers have become considerably more efficient

the asymptotic variance of the univariate pls estimator in this short note we derive an expression for the asymptotic covariance matrix of the univariate partial least squares pls estimator in contrast to m c denham j chemometrics 11 1997 39 who provided a locally linear approximation based on a recursive definition of the estimator we derive a more compact expression for the asymptotic covariance matrix by combining a standard convergence result with matrix differential calculus in particular the approach of j r magnus and h neudecker matrix differential calculus with applications in statistics and econometrics revised ed wiley chichester uk 1991 we also describe some theoretical and practical aspects of calculating the asymptotic covariance matrix and illustrate its use on spectroscopic data

the second generation shifted boundary method and its numerical analysis abstract recently the shifted boundary method sbm was proposed within the class of unfitted or immersed or embedded finite element methods by reformulating the original boundary value problem over a surrogate approximate computational domain the sbm avoids integration over cut cells and the associated problematic issues regarding numerical stability and matrix conditioning accuracy is maintained by modifying the original boundary conditions using taylor expansions hence the name of the method that shifts the location and values of the boundary conditions in this article we present enhanced variational sbm formulations for the poisson and stokes problems with improved flexibility and robustness these simplified variational forms allow to relax some of the assumptions required by the mathematical proofs of stability and convergence of earlier implementations first we show that these new sbm implementations can be proved asymptotically stable and convergent even without the rather restrictive assumption that the inner product between the normals to the true and surrogate boundaries is positive second we show that it is not necessary to introduce a stabilization term involving the tangential derivatives of the solution at dirichlet boundaries therefore avoiding the calibration of an additional stabilization parameter finally we prove enhanced l 2 estimates without the cumbersome assumption of earlier proofs that the surrogate domain is convex instead we rely on a conventional assumption that the boundary of the true domain is smooth which can also be replaced by requiring convexity of the true domain the aforementioned improvements open the way to a more general and efficient implementation of the shifted boundary method particularly in complex three dimensional geometries we complement these theoretical developments with numerical experiments in two and three dimensions

isogeometric analysis for trimmed cad surfaces in this work an isogeometric analysis for trimmed cad surfaces in 2d linear elasticity problem is presented the main benefit of the proposed method is that no additional modeling for the analysis of the trimmed surface is necessary as a pioneering attempt to deal with the trimmed surfaces in isogeometric analysis the information on the trimming curves and trimmed surfaces exported from cad system is directly utilized for analysis for this trimmed elements are searched through using a projection scheme for the integration of trimmed elements nurbs enhanced integration scheme employed in nefem is adopted for the isogeometric analysis the construction of the stiffness matrix based on the spline basis function is presented in the formulation the information on the trimming curves is used not only for obtaining integration points but also for calculating the jacobian through error analyses of the verification examples the robustness and effectiveness of the proposed method are investigated it is observed that the proposed method gives the theoretical convergence rate multiple trimming curve problems which are difficult to analyze with conventional isogeometric analysis are easily treated with the proposed method an illuminating example is given in the same example t spline local refinement is also employed

an approach to refining three dimensional tetrahedral meshes based on delaunay transformations a technique for refining three dimensional tetrahedral meshes is proposed in this paper the proposed technique is capable of treating arbitrary unstructured tetrahedral meshes convex or non convex with multiple regions resulting in high quality constrained delaunay triangulations the tetrahedra generated are of high quality nearly equilateral sliver tetrahedra which present a real problem to many algorithms are not produced with the new method the key to the generation of high quality tetrahedra is the iterative application of a set of topological transformations based on the voronoi delaunay theory and a reposition of nodes technique the computational requirements of the proposed technique are in linear relationship with the number of nodes and tetrahedra making it ideal for direct employment in a fully automatic finite element analysis system for 3 d adaptive mesh refinement application to some test problems is presented to show the effectiveness and applicability of the new method

weighted extended b spline approximation of dirichlet problems we describe a new finite element method which uses weighted extended b splines on a regular grid as basis functions for solving dirichlet problems on bounded domains in arbitrary dimensions this web method does not require any grid generation and can be implemented very efficiently it yields smooth high order accurate approximations with relatively low dimensional subspaces

bouncing ball orbits and symmetry breaking effects in a three dimensional chaotic billiard we study the classical and quantum mechanics of a three dimensional stadium billiard it consists of two quarter cylinders that are rotated with respect to each other by 90 ifmmode circ else textdegree fi and it is classically chaotic the billiard exhibits only a few families of nongeneric periodic orbits we introduce an analytic method for their treatment the length spectrum can be understood in terms of the nongeneric and unstable periodic orbits for unequal radii of the quarter cylinders the level statistics agree well with predictions from random matrix theory for equal radii the billiard exhibits an additional symmetry we investigated the effects of symmetry breaking on spectral properties moreover for equal radii we observe a small deviation of the level statistics from random matrix theory this led to the discovery of stable and marginally stable orbits which are absent for unequal radii

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

finite element methods of least squares type we consider the application of least squares variational principles to the numerical solution of partial differential equations our main focus is on the development of least squares finite element methods for elliptic boundary value problems arising in fields such as fluid flows linear elasticity and convection diffusion for many of these problems least squares principles offer numerous theoretical and computational advantages in the algorithmic design and implementation of corresponding finite element methods that are not present in standard galerkin discretizations most notably the use of least squares principles leads to symmetric and positive definite algebraic problems and allows us to circumvent stability conditions such as the inf sup condition arising in mixed methods for the stokes and navier stokes equations as a result application of least squares principles has led to the development of robust and efficient finite element methods for a large class of problems of practical importance

the role of bezier extraction in adaptive isogeometric analysis local refinement and hierarchical refinement summary we present two adaptive refinement techniques namely adaptive local refinement and adaptive hierarchical refinement for isogeometric analysis an element wise point of view is adopted exploiting bezier extraction which facilitates the implementation of adaptive refinement in isogeometric analysis locally refined t splines lr t splines and hierarchical t splines are employed for the description of the geometry as well as for the approximation of the solution space in the analysis the refinement is carried out with the aid of a subdivision operator which is computed by again exploiting the bezier extraction operator the concept and algorithm of an element based adaptive isogeometric analysis are illustrated numerical examples are given to examine the accuracy the convergence and the condition number this article is protected by copyright all rights reserved

exact imposition of inhomogeneous dirichlet boundary conditions based on weighted finite cell method and level set function abstract the imposition of inhomogeneous dirichlet boundary conditions idbcs is essential in numerical analysis of a structure it is especially difficult and no longer straightforward whenever non conformal mesh is used to discretize a structure the original contribution of this paper is to develop a weighted finite cell method fcm with high computing accuracy the commonly used transfinite interpolation in the computer aided design cad community is now extended to define boundary value function so that the idbcs are imposed exactly unlike existing weak imposition forms and weighted interpolation methods it is shown that the weighting function and boundary value function involved in the weighted fcm can directly be applied to the physical field independently of the mesh discretization besides level set function lsf is used to represent the geometry of the considered physical domain to verify the effectiveness convergence performance and the generality of the proposed method numerical examples ranging from purely elastic to thermoelastic problems are illustrated computing results are compared with analytical and fea solutions

imposition of essential boundary conditions in the material point method summary there is increasing interest in the material point method mpm as a means of modelling solid mechanics problems in which very large deformations occur e g in the study of landslides and metal forming however some aspects vital to wider use of the method have to date been ignored in particular methods for imposing essential boundary conditions in the case where the problem domain boundary does not coincide with the background grid element edges in this paper we develop a simple procedure originally devised for standard finite elements for the imposition of essential boundary conditions to the mpm expanding its capabilities to boundaries of any inclination to the authors knowledge this is the first time that a method has been proposed that allows arbitrary dirichlet boundary conditions zero and non zero values at any inclination to be imposed in the mpm the method presented in this paper is different from other mpm boundary approximation approaches in that i the boundaries are independent of the background mesh ii artificially stiff regions of material points are avoided and iii the method does not rely on spurious mirroring of the problem domain to imposed symmetry the main contribution of this work is equally applicable to standard finite elements and the mpm this article is protected by copyright all rights reserved

truncated second main theorem with moving targets we prove a truncated second main theorem for holomorphic curves intersecting a finite set of moving or fixed hyperplanes the set of hyperplanes is assumed to be non degenerate previously only general position or subgeneral position was considered

the shifted boundary method for embedded domain computations part i poisson and stokes problems abstract we propose a new finite element method for embedded domain computations which falls in the category of surrogate approximate boundary algorithms the key feature of the proposed approach is the idea of shifting the location where boundary conditions are applied from the true to the surrogate boundary and to appropriately modify the shifted boundary conditions enforced weakly in order to preserve optimal convergence rates of the numerical solution this process yields a method which in our view is simple efficient and also robust since it is not affected by the small cut cell problem although general in nature here we apply this new concept to the poisson and stokes problems we present in particular the full analysis of stability and convergence for the case of the poisson operator and numerical tests for both the poisson and stokes equations for geometries of progressively higher complexity

shape optimization of dirichlet boundaries based on weighted b spline finite cell method and level set function abstract this paper addresses extended shape optimization problems where structural supports i e dirichlet boundaries and free boundaries are simultaneously optimized unlike traditional fem the weighted b spline finite cell method fcm is applied as structural analysis tool and combined with the level set function lsf to take into account dirichlet boundary condition dbc automatically through penalization of the displacement field the proposed shape optimization method achieves a comprehensive integration of fixed grid b spline shape function and level set function as both the structure and dirichlet boundaries are described in the form of lsf any modification of dirichlet boundaries can be made in a straightforward way as easily as free boundaries by changing continuous design variables meanwhile the computing accuracy is ensured within the framework of fixed grid owing to the quadtree refinements of boundary cells stress related shape optimization problems are finally solved to demonstrate the merit and validity of the proposed optimization method in dealing with shape optimization of dirichlet boundaries

isogeometric analysis on implicit domains using weighted extended pht splines abstract the urging quest for diminishing the gap between finite element methods and computer aided design is growing every day isogeometric analysis is driving that search a step forward in contrast with traditional finite element methods one of the areas that need more investigation is using isogeometric analysis on domains that contain singularities in this case the parameterization process is occasionally non trivial and some of the inherited basis functions are not well defined producing rough solutions moreover nurbs basis functions are not the best choice to perform local refinement since adaptivity is indispensable in this study a new approach that serves as a generalization for some of the ideas in both the isogeometric analysis and the web method is introduced isogeometric analysis on implicit domains replaces mesh generation with implicit spline weight function generation and construct weighted extended pht splines for analysis the geometry and the analysis both are built starting from the same function space avoiding parameterization the pht splines are adaptive and allow local refinement naturally leading to a higher accuracy while dealing with singularities an illustration with numerical examples demonstrates the efficiency and performance of the proposed method

transfinite mean value interpolation in general dimension mean value interpolation is a simple fast linearly precise method of smoothly interpolating a function given on the boundary of a domain for planar domains several properties of the interpolant were established in a recent paper by dyken and the second author including sufficient conditions on the boundary to guarantee interpolation for continuous data a formula for the normal derivative at the boundary and the construction of a hermite interpolant when normal derivative data is also available in this paper we generalize these results to domains in arbitrary dimension

an unfitted hp adaptive finite element method based on hierarchical b splines for interface problems of complex geometry generating finite element discretizations with direct interface parameterizations constitutes a considerable computational expense in case of complex interface geometries the paper at hand introduces a b spline finite element method which circumvents parameterization of interfaces and offers fast and easy meshing irrespective of the geometric complexity involved its core idea is the adaptive approximation of discontinuities by hierarchical grid refinement which adds several levels of local basis functions in the close vicinity of interfaces but unfitted to their exact location so that a simple regular grid of knot span elements can be maintained numerical experiments show that an hp refinement strategy which simultaneously increases the polynomial degree of the b spline basis and the levels of refinement around interfaces achieves exponential rates of convergence despite the presence of discontinuities it is also demonstrated that the hierarchical b spline fem can be used to transfer the recently introduced finite cell concept to geometrically nonlinear problems its computational performance imposition of unfitted boundary conditions and fast hierarchical grid generation are illustrated for a set of benchmark problems in one two and three dimensions and the advantages of the regular grid approach for complex geometries are demonstrated by the geometrically nonlinear simulation of a voxel based foam composite

a statistical view of some chemometrics regression tools chemometrics is a field of chemistry that studies the application of statistical methods to chemical data analysis in addition to borrowing many techniques from the statistics and engineering literatures chemometrics itself has given rise to several new data analytical methods this article examines two methods commonly used in chemometrics for predictive modeling partial least squares and principal components regression from a statistical perspective the goal is to try to understand their apparent successes and in what situations they can be expected to work well and to compare them with other statistical methods intended for those situations these methods include ordinary least squares variable subset selection and ridge regression

adaptive multiresolution refinement with distance fields this paper describes a multiresolution approach to field modeling that can be used with any meshfree or mesh based method for adaptive solution refinement the refined solution is represented as a superposition of a coarse unrefined solution and a sequence of refinements that provide additional degrees of freedom with higher spatial or functional resolution each refinement is treated as a solution to a boundary value problem within a specified refinement window the proposed approach is based on the meshfree method with distance fields comput mech 2000 25 305 316 eng comput 2002 18 4 295 311 and guarantees cm continuity of the refined solutions with matching or non matching grids the method does not restrict the shape of the refinement window and does not place any constraints on the type of basis functions or relative position and resolution of the refinement grids combining the proposed approach with hierarchical space decompositions and a posteriori error estimators results in an effective tool for automatic solution refinement carefully chosen numerical examples illustrate the power and advantages of the proposed approach copyright 2007 john wiley sons ltd

an implicit boundary finite element method with extension to frictional sliding boundary conditions and elasto plastic analyses abstract implicit boundary methods which enrich the interpolation structure with implicit weight functions are straightforward methods for the enforcement of dirichlet boundary conditions in this article we follow the implicit boundary method that uses approximate step functions the step boundary method developed by kumar et al and provide modifications that have several advantages roller boundary conditions have wide practical applications in engineering however the step boundary method for roller boundary conditions with inclinations has yet to be fully formulated through to the final linear system of equations thus we provide a complete derivation that leads to simplified stiffness matrices compared to the original approach which can be implemented directly in fictitious domain finite element analysis the approach is then extended we believe for the first time to the nonlinear cases of frictional boundary conditions and elasto plastic material behaviour the proposed formulation and procedures are validated on a number of example problems that test different aspects of the method

the shifted boundary method for embedded domain computations part ii linear advection diffusion and incompressible navier stokes equations abstract we propose a new embedded finite element method for the linear advection diffusion equation and the laminar and turbulent incompressible navier stokes equations the proposed method belongs to the class of surrogate approximate boundary algorithms and is based on the idea of shifting the location where boundary conditions are applied from the true to a surrogate boundary accordingly boundary conditions enforced weakly are appropriately modified to preserve optimal error convergence rates we include the full analysis of stability and convergence of the method in the linear advection diffusion equation and a battery of tests for the case of laminar and turbulent incompressible navier stokes equations we also discuss the conservation properties of the proposed method in all cases

subdivision stabilised immersed b spline finite elements for moving boundary flows an immersed finite element method is presented to compute flows with complex moving boundaries on a fixed cartesian grid the viscous incompressible fluid flow equations are discretized with b spline basis functions the two scale relation for b splines is used to implement an elegant and efficient technique to satisfy the lbb condition on non grid aligned fluid domains and at moving boundaries the boundary conditions are enforced with a consistent penalty method as originally proposed by nitsche in addition a special extrapolation technique is employed to prevent the loss of numerical stability in presence of arbitrarily small cut cells the versatility and accuracy of the proposed approach is demonstrated by means of convergence studies and comparisons with previous experimental and computational investigations

a fictitious domain distributed lagrange multiplier based fluid structure interaction scheme with hierarchical b spline grids abstract we present a numerical scheme for fluid structure interaction based on hierarchical b spline grids and fictitious domain distributed lagrange multipliers the incompressible navier stokes equations are solved over a cartesian grid discretised with b splines the fluid grid near the immersed solids is refined locally using hierarchical b splines the immersed solid is modelled as geometrically exact beam discretised with standard linear lagrange shape functions the kinematic constraint at the fluid solid interface is enforced with distributed lagrange multipliers the unconditionally stable and second order accurate generalised method is used for integration in time for both the fluid and solid domains a fully implicit and fully coupled solution scheme is developed by using the newton raphson method to solve the non linear system of equations obtained with galerkin weak formulation first the spatial and temporal convergence of the proposed scheme is assessed by studying steady and unsteady flow past a fixed cylinder then the scheme is applied to several benchmark problems to demonstrate the efficiency and robustness of the proposed scheme the results obtained with the present scheme are compared with the reference values

multigrid methods with web splines we describe and analyze a multigrid algorithm for finite element approximations of second order elliptic boundary value problems with weightedextended b splines web splines this new technique provides high accuracy with relatively low dimensional subspaces does not require any grid generation and is ideally suited for hierarchical solution techniques in particular we show that the standard w cycle yields uniform convergence i e the required number of iterations is bounded independent of the grid width

stable isogeometric analysis of trimmed geometries abstract we explore extended b splines as a stable basis for isogeometric analysis with trimmed parameter spaces the stabilization is accomplished by an appropriate substitution of b splines that may lead to ill conditioned system matrices the construction for non uniform knot vectors is presented the properties of extended b splines are examined in the context of interpolation potential and linear elasticity problems and excellent results are attained the analysis is performed by an isogeometric boundary element formulation using collocation it is argued that extended b splines provide a flexible and simple stabilization scheme which ideally suits the isogeometric paradigm

metric based upscaling we consider divergence form elliptic operators in dimension n ge 2 with l coefficients although solutions of these operators are only holder continuous we show that they are differentiable c1 with respect to harmonic coordinates it follows that numerical homogenization can be extended to situations where the medium has no ergodicity at small scales and is characterized by a continuum of scales this new numerical homogenization method is based on the transfer of a new metric in addition to traditional averaged homogenized quantities from subgrid scales into computational scales error bounds can be given and this method can also be used as a compression tool for differential operators 2006 wiley periodicals inc

a note on the stability parameter in nitsche s method for unfitted boundary value problems abstract nitsche s method is a popular approach to implement dirichlet type boundary conditions in situations where a strong imposition is either inconvenient or simply not feasible the method is widely applied in the context of unfitted finite element methods of the classical symmetric nitsche s method it is well known that the stabilization parameter in the method has to be chosen sufficiently large to obtain unique solvability of discrete systems in this short note we discuss an often used strategy to set the stabilization parameter and describe a possible problem that can arise from this we show that in specific situations error bounds can deteriorate and give examples of computations where nitsche s method yields large and even diverging discretization errors

powell sabin splines with boundary conditions for polygonal and non polygonal domains powell sabin splines are piecewise quadratic polynomials with a global c 1 continuity defined on conforming triangulations imposing boundary conditions on such a spline leads to a set of constraints on the spline coefficients first we discuss boundary conditions defined on a polygonal domain before we treat boundary conditions on a general curved domain boundary we consider dirichlet and neumann conditions and we show that a particular choice of the ps triangles at the boundary can greatly simplify the corresponding constraints finally we consider an application where the techniques developed in this paper are used the numerical solution of a partial differential equation by the galerkin and collocation method

analysis of the shifted boundary method for the stokes problem abstract the analysis of stability and accuracy of the shifted boundary method is developed for the stokes flow equations the key feature of the shifted boundary method an embedded finite element method is the shifting of the location where boundary conditions are applied from the true to a surrogate boundary and an appropriate modification shifting of the value of the boundary conditions an inf sup condition is proved for the variational formulation associated to the shifted boundary method and we derive by way of strang s second lemma an optimal error estimate in the natural sbm norm we also derive an l 2 error estimate for the velocity field by means of an extension of the aubin nitsche approach to embedded non consistent mixed finite element methods

a local solution approach for adaptive hierarchical refinement in isogeometric analysis abstract adaptively refining the state variables and fast solving the equations are two key issues in isogeometric analysis currently existing research more focuses on the first issue while the efficiency issue is not elaborated when the overall equations are solved at each refinement step this article develops an approach to promote the efficiency by locally solving the equations that govern the refined variables at each refinement step here hierarchical refinement of the state variables in nurbs is adopted and transformational matrix that transforms the basis functions from the current level to the refined level is utilized to handle the relations between the equations at different levels by dividing the coefficient matrix into blocks and estimating the error of state variables for different blocks we represent the global solution method in a hierarchical structure and separately construct and solve equations to obtain the solution for each level in the structure we validate the proposed method by some steady heat transfer and structural analysis problems in domains with different dimensionality we have compared the numerical result accuracy as well as the time cost of the proposed method with those of current available refinement approaches and it turns out to be very promising

a stabilised immersed boundary method on hierarchical b spline grids abstract in this work an immersed boundary finite element method is proposed which is based on a hierarchically refined cartesian b spline grid and employs the non symmetric and penalty free version of nitsche s method to enforce the boundary conditions the strategy allows for h and p refinement and employs a so called ghost penalty term to stabilise the cut cells an effective procedure based on hierarchical subdivision and sub cell merging which avoids excessive numbers of quadrature points is used for the integration of the cut cells a basic laplace problem is used to demonstrate the effectiveness of the cut cell stabilisation and of the penalty free nitsche method as well as their impact on accuracy the methodology is also applied to the incompressible navier stokes equations where the supg pspg stabilisation is employed simulations of the lid driven cavity flow and the flow around a cylinder at low reynolds number show the good performance of the methodology excessive ill conditioning of the system matrix is robustly avoided without jeopardising the accuracy at the immersed boundaries or in the field

the shifted boundary method for hyperbolic systems embedded domain computations of linear waves and shallow water flows abstract we propose a new computational approach for embedded boundary simulations of hyperbolic systems and in particular the linear wave equations and the nonlinear shallow water equations the proposed approach belongs to the class of surrogate approximate boundary algorithms and is based on the idea of shifting the location where boundary conditions are applied from the true to a surrogate boundary accordingly boundary conditions enforced weakly are appropriately modified to preserve optimal error convergence rates this framework is applied here in the setting of a stabilized finite element method even though other spatial discretization techniques could have been employed accuracy stability and robustness of the proposed method are tested by means of an extensive set of computational experiments for the acoustic wave propagation equations and shallow water equations comparisons with standard weak boundary conditions imposed on body fitted grids which conform to the geometry of the computational domain boundaries are also presented

radially symmetric weighted extended b spline model in this paper the weighted extended basis splines approach in the finite element method is applied to the electrostatic electromagnetic wave and bioheat problems for inhomogeneous boundary conditions and radially symmetric structures this new method which does not need mesh generation overcomes some of the drawbacks of using meshes and piecewise uniform or linear trial functions two dimensional radially symmetric electrostatic and electromagnetic wave equations are evaluated we also attempt to propose a three dimensional radially symmetric unexposed human eye model for simulating changes in corneal temperature using these new finite elements in conjunction with linear quadratic and cubic b splines our findings indicate that weighted extended basis spline solutions improve the standard finite element method the simulation results which are verified using the values reported in the literature point out to better efficiency in terms of the accuracy level

weak impositions of dirichlet boundary conditions in solid mechanics a critique of current approaches and extension to partially prescribed boundaries abstract in this article we first review various approaches developed to date for the weak imposition of dirichlet boundary conditions in fictitious domain analysis for elasticity problems the hellinger reissner h r principle the linked lagrange multiplier llm method the implicit boundary method and the fat boundary method are discussed along with the well known lagrange multiplier penalty and nitsche s methods we state these approaches in a common form starting with energy functionals and weak forms and discretise using the fictitious domain finite element method previous formulations of these methods were in general developed for full prescription along the dirichlet boundary which generally implies no local effect of boundary inclination however partially prescribed conditions such as the structural roller boundary condition with inclination have wide practical applications in engineering here we provide techniques of imposing such boundary conditions in these methods in detail for those methods that contain algorithmic parameters such as the penalty and nitsche s methods extra computation or empirical estimation is necessary to decide values of the parameters and hence we discuss parametric and convergence behaviours through numerical examples to provide guidance on the choice of parameters

mass lumping techniques in the spectral element method on the equivalence of the row sum nodal quadrature and diagonal scaling methods abstract in the context of wave propagation analysis the spectral element method sem in conjunction with a diagonal mass matrix is often the method of choice therefore it is of high importance to investigate the influence of different mass lumping schemes on the accuracy of the numerical results to this end we compare the performance of three established methods including the row sum method the nodal quadrature method and the diagonal scaling method the theoretical analysis of these methods reveals a close connection between them under certain conditions that are discussed in detail in this article we are able to show a direct equivalence between these three approaches in this regard the attainable accuracy of the numerical integration of the mass matrix plays an important role by means of several dynamic benchmark problems we verify the theoretical results and illustrate the convergence properties of the lumped mass matrix sem in comparison to a formulation based on the consistent mass matrix

meshless local b spline fd method and its application for 2d heat conduction problems with spatially varying thermal conductivity abstract in this paper a new class of meshless methods based on local collocation and b spline basis functions is presented for solving elliptic problems the proposed approach is called as meshless local b spline basis functions based finite difference local b fd method the method was straightforward to develop and program as it was truly meshless only scattered nodal distribution was required hence avoiding at all mesh connectivity for field variable approximation and integration in the method any governing equations were discretized by b spline approximation in the spirit of fd technique using local b spline collocation i e any derivative at a point or node was stated as neighboring nodal values based on the b spline interpolants in addition as b spline basis functions pose favorable properties such as i easy to construct to any arbitrary order degree ii have partition of unity property and iii can be easily designed to pose the kronecker delta property the shape function construction as well as the imposition of boundary conditions can be incorporated efficiently in the present method the applicability and capability of the present local b fd method were demonstrated through several heat conduction problems with heat generation and spatially varying conductivity

implicit boundary method for analysis using uniform b spline basis and structured grid several analysis techniques such as extended finite element method x fem have been developed recently which use structured grid for the analysis implicit boundary method uses implicit equations of the boundary to apply boundary conditions in x fem framework using structured grids solution structures for test and trial functions are constructed using implicit equations such that the boundary conditions are satisfied even if there are no nodes on the boundary in this paper this method is applied for analysis using uniform b spline basis defined over a structured grid solution structures that are c1 or c2 continuous throughout the analysis domain can be constructed using b spline basis functions as a structured grid does not conform to the geometry of the analysis domain the boundaries of the analysis domain are defined independently using equations of the boundary curves surfaces compared with conforming mesh it is easier to generate structured grids that overlap the geometry and the elements in the grid are regular shaped and undistorted numerical examples are presented to demonstrate the performance of these b spline elements the results are compared with analytical solutions as well as with traditional finite element solutions convergence studies for several examples show that b spline elements provide accurate solutions with fewer elements and nodes compared with traditional fem they also provide continuous stress and strain in the analysis domain thus eliminating the need for smoothing stress strain results copyright 2008 john wiley sons ltd

dual and approximate dual basis functions for b splines and nurbs comparison and application for an efficient coupling of patches with the isogeometric mortar method abstract this contribution defines and compares different methods for the computation of dual basis functions for b splines and non uniform rational b splines nurbs they are intended to be used as test functions for the isogeometric mortar method but other fields of application are possible too three different concepts are presented and compared the first concept is the explicit formula for the computation of dual basis functions for nurbs proposed in the work of carl de boor these dual basis functions entail minimal support i e the support of the dual basis functions is equal to the support of the corresponding b spline basis functions in the second concept dual basis functions are derived from the inversion of the gram matrix these dual basis functions have global support along the interface the third concept is the use of approximate dual basis functions which were initially proposed for the use in harmonic analysis the support of these functions is local but larger than the support of the associated b spline basis functions we propose an extension of the approximate dual basis functions for nurbs basis functions after providing the general formulas we elaborate explicit expressions for several degrees of spline basis functions all three approaches are applied in the frame of the mortar method for the coupling of non conforming nurbs patches a method which allows complex discretizations with multiple intersecting interfaces is presented numerical examples show that the explicitly defined dual basis functions with minimal support severely deteriorate the global stress convergence behavior of the mechanical analysis this fact is in accordance with mathematical findings in literature which state that the optimal reproduction degree of arbitrary functions is not possible without extending the support of the dual basis functions the dual basis functions computed from the inverse of the gram matrix yield accurate numerical results but the global support yields significantly higher computational costs in comparison to computations of conforming meshes only the approximate dual basis functions yield accurate and efficient computations where neither accuracy nor efficiency is significantly deteriorated in comparison to computations of conforming meshes all basic cases of t intersections and star intersections are studied furthermore an example which combines all basic cases in a complex discretization is given the applicability of the presented method for the nonlinear case and for shell formulations is shown with the help of one numerical example

an optimal order process for solving finite element equations a k level iterative procedure for solving the algebraic equations which arise from the finite element approximation of elliptic boundary value problems is presented and analyzed the work estimate for this procedure is proportional to the number of unknowns an optimal order result general geometry is permitted for the underlying domain but the shape plays a role in the rate of convergence through elliptic regularity finally a short discussion of the use of this method for parabolic problems is presented

weighted extended b spline method for the approximation of the stationary stokes problem a new stabilized mesh free method for the approximation of the stokes problem using weighted extended b splines web splines as shape functions has been proposed the web spline based bilinear velocity constant pressure element satisfies the so called inf sup condition or ladyshenskaya babus ka brezzi lbb condition the main advantage of this method over standard finite element methods is that it uses regular grids instead of irregular partitions of domain thus eliminating the difficult and time consuming pre processing step convergence and condition number estimates are derived numerical experiments in two space dimensions confirm the theoretical predictions

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

initial solution estimation for one step inverse isogeometric analysis in sheet metal stamping abstract recently isogeometric analysis iga based on incremental methods for simulating the stamping process has been researched to the best of our knowledge however few studies have combined iga and one step inverse approach which is based on total deformation theory of plasticity a key step for one step inverse iga is to estimate a good initial solution traditional mesh based initial solution algorithms for one step inverse approach are not suitable for one step inverse nurbs based iga in this paper we presented a method which can rapidly unfold the undevelopable nurbs surface onto a planar domain and obtain a good initial solution estimation for one step inverse iga the key idea of the presented method is unfolding the control net of a nurbs surface for isogeometric analysis by energy based initial solution estimation algorithm in addition we developed a cutting stitching algorithm which can separate a complex control net into several parts with simple shapes numerical examples illustrate the initial solutions using the presented method are approaching the final results by one step inverse finite element method this implies that the iterative steps and computational time of one step inverse iga will be reduced significantly compared with that of one step inverse finite element method

b splines with applications to isogeometric analysis abstract a novel spline named b splines boundary plus plus splines is developed to address the expression of a trimmed nurbs patch in an analytic form which is a central idea of surface representation the presented method converts each trimmed nurbs patch into a b spline patch that incorporates of specific boundary points as the boundary presentation emphasis is placed on the construction of a new analytic formula of a trimmed nurbs patch defined by the boundary points at the trimming curves and a group of enriched control points b spline basis functions are linearly independent build a partition of unity and satisfy the kronecker delta property each b spline basis function is a linear combination of the basis functions of the trimmed nurbs patch these properties allow imposing the dirichlet boundary conditions strongly at the boundary of the trimmed patch without the necessity of modifying the basis functions of the trimmed patch isogeometric analysis using b splines for two dimensional elastic solids is also proposed several numerical examples are used to demonstrate the reliability of the presented method the numerical example for the patch test illustrates that the b spline patch passes the standard patch test

condition number analysis and preconditioning of the finite cell method the isogeometric finite cell method in which a domain is immersed in a structured background mesh suffers from conditioning problems when cells with small volume fractions occur in this contribution we establish a rigorous scaling relation between the condition number of i fcm system matrices and the smallest cell volume fraction ill conditioning stems either from basis functions being small on cells with small volume fractions or from basis functions being nearly linearly dependent on such cells based on these two sources of ill conditioning an algebraic preconditioning technique is developed which is referred to as symmetric incomplete permuted inverse cholesky sipic a detailed numerical investigation of the effectivity of the sipic preconditioner in improving i fcm condition numbers and in improving the convergence speed and accuracy of iterative solvers is presented for the poisson problem and for two and three dimensional problems in linear elasticity in which nitche s method is applied in either the normal or tangential direction the accuracy of the preconditioned iterative solver enables mesh convergence studies of the finite cell method

on cartan s theorem and cartan s conjecture we give a mild generalization of cartan s theorem on value distribution for a holomorphic curve in projective space relative to hyperplanes this generalization is used to complete the proof of the following theorem claimed in an earlier paper by the author given hyperplanes in projective space in general position there exists a finite union of proper linear subspaces such that all holomorphic curves not contained in that union even linearly degenerate curves satisfy the inequality of cartan s theorem except for the ramification term in addition it is shown how these methods can lead to a shorter proof of nochka s theorem on cartan s conjecture and in the number field case how nochka s theorem gives a short proof of wirsing s theorem on approximation of algebraic numbers by algebraic numbers of bounded degree