orthogonal grid generation of an irregular region using a local polynomial collocation method abstract in this study a 2 d orthogonal grid generation model is developed by solving the governing equations of coordinate transformation with a local polynomial collocation method accompanied with the moving least squares mls approach this method was developed in a way that on the boundaries both the governing equation and boundary condition are satisfied so it is more robust and accurate than conventional collocation methods though the method used to solve the coordinate transforming equations is meshless it does not deteriorate the value of present work because most numerical models in modern use are grid dependent and grid generation of service to these models is still strongly desired particularly for finite difference models in irregular domains before applying to grid generation problems the performance of present method is tested by a bench mark potential flow problem additional to two basic grid generation problems a bottleneck problem of previous works which contains zero degree corners in the domain is carried out finally the model is applied to the orthogonal grid generation in a multi connected domain the correctness is testified by checking the orthogonality of the generated results

isogeometric analysis using t splines we explore t splines a generalization of nurbs enabling local refinement as a basis for isogeometric analysis we review t splines as a surface design methodology and then develop it for engineering analysis applications we test t splines on some elementary two dimensional and three dimensional fluid and structural analysis problems and attain good results in all cases we summarize the current status of t splines their limitations and future possibilities

rotation free isogeometric thin shell analysis using pht splines this paper presents a novel approach for isogeometric analysis of thin shells using polynomial splines over hierarchical t meshes pht splines the method exploits the flexibility of t meshes for local refinement the main advantage of the pht splines in the context of thin shell theory is that it achieves c1 continuity so the kirchhoff love theory can be used in pristine form no rotational degrees of freedom are needed numerical results show the excellent performance of the present method

isogeometric analysis using polynomial splines over hierarchical t meshes for two dimensional elastic solids isogeometric analysis has become a powerful alternative to standard finite elements due to its flexibility in handling complex geometries one of the major drawbacks of nurbs based isogeometric finite elements is the inefficiency of local refinement in this study we present an alternative to nurbs based isogeometric analysis that allows for local refinement the idea is based on polynomial splines and exploits the flexibility of t meshes for local refinement the shape functions satisfy important properties such as non negativity local support and partition of unity several numerical examples are used to demonstrate the reliability of the present method

unique normal form property of compatible term rewriting systems a new proof of chew s theorem abstract we present a new and complete proof of chew s theorem which states that a compatible term rewriting system has the unique normal form property i e a b implies a b for any normal forms a b

an isogeometric analysis approach for the study of the gyrokinetic quasi neutrality equation in this work a new discretization scheme for the gyrokinetic quasi neutrality equation is proposed it is based on isogeometric analysis the iga which relies on nurbs functions accommodates arbitrary coordinates and the use of complicated computation domains moreover arbitrary high order degree of basis functions can be used and their regularity enables the use of a low number of elements here this approach is successfully tested on elliptic problems like the quasi neutrality equation arising in gyrokinetic models in this last application when polar coordinates are considered a fast solver can be used and the non locality is dealt with a suitable decomposition which reduces the resolution of the gyrokinetic quasi neutrality equation to a sequence of local 2d elliptic problems

an improved nearly orthogonal structured mesh generation system with smoothness control functions this paper presents an improved nearly orthogonal structured mesh generation system with a set of smoothness control functions which were derived based on the ratio between the jacobian of the transformation matrix and the jacobian of the metric tensor the proposed smoothness control functions are capable of relaxing the local strong orthogonal conditions so that nearly orthogonal but smooth mesh can be achieved examples and applications are also investigated in this paper to demonstrate the effects of the proposed mesh generation system

h1 parametrizations of complex planar physical domains in isogeometric analysis abstract isogeometric analysis is a method for solving geometric partial differential equations pdes generating parametrizations of a pde s physical domain is a basic and important issue within isogeometric analysis framework in this paper we present a global h 1 parametrization method for a complex planar physical domain

2d nearly orthogonal mesh generation with controls on distortion function a method to control the distortion function of the ryskin and leal rl orthogonal mesh generation system is presented the proposed method considers the effects from not only the local orthogonal condition but also the local smoothness condition the geometry and the mesh size on the distortion function the distortion function is determined by both the scale factors and the averaged scale factors of the constant mesh lines two adjustable parameters are used to control the local balance of the orthogonality and the smoothness the proposed method is successfully applied to several benchmark examples and the natural river channels with complex geometries

a fully locking free isogeometric approach for plane linear elasticity problems a stream function formulation in this paper we study plane incompressible elastic problems by means of a stream function formulation such that a divergence free displacement field can be computed from a scalar potential the numerical scheme is constructed within the framework of nurbs based isogeometric analysis and we take advantage of the high continuity guaranteed by nurbs basis functions in order to obtain the displacement field from the potential differentiation as a consequence the obtained numerical scheme is automatically locking free in the presence of the incompressibility constraint a discontinuous galerkin approach is proposed to deal with multiple mapped possibly multiply connected domains extensive numerical results are provided to show the method capabilities 2007 elsevier b v all rights reserved

isogeometric analysis of free surface flow abstract this paper presents the first application of isogeometric analysis a new computational technology built on higher order and higher continuity basis functions employed in computer aided design and computer graphics to the computation of free surface phenomena described using the level set approach the method is based on the variational framework that is suitable for discretization by standard finite elements as well as the basis functions employed in isogeometric analysis the underlying numerical formulation globally conserves mass and preserves a sharp air water interface for the entire length of the simulation the numerical tests indicate that the proposed methodology gives an accurate description of the free surface behavior in both quasi steady and dynamic regimes furthermore very good per degree of freedom accuracy is obtained when higher order and higher continuity isogeometric discretizations are employed in free surface computations

a h1 integrability condition of surfaces with singular parametrizations in isogeometric analysis abstract singularities of a surface s geometric mapping or parametrization are often unavoidable especially when a complex surface is considered in isogeometric analysis singularities impact the regularity of test functions when a second order partial differential equation such as a laplace beltrami equation is solved the test functions should satisfy the h 1 regularity which may be destroyed by singularities in this paper we consider the h 1 regularity of test functions on a surface by parametrization with isolated singularities a h 1 integrability condition is presented and we apply this condition to discuss the h 1 regularity of test functions by two common singular parametrizations of surfaces such as spheres and d patches

unique normal forms for lambda calculus with surjective pairing we consider the equational theory of calculus extended with constants 0 1 and axioms for surjective pairing 0 xy x 1 xy y 0x 1x x two reduction systems yielding the equality of are introduced the first is not confluent and for the second confluence is an open problem it is shown however that in both systems each term possessing a normal form has a unique normal form some additional properties and problems in the syntactical analysis of and the corresponding reduction systems are discussed

isogeometric analysis for surface pdes with extended loop subdivision abstract we investigate the isogeometric analysis for surface pdes based on the extended loop subdivision approach the basis functions consisting of quartic box splines corresponding to each subdivided control mesh are utilized to represent the geometry exactly and construct the solution space for dependent variables as well which is consistent with the concept of isogeometric analysis the subdivision process is equivalent to the h refinement of nurbs based isogeometric analysis the performance of the proposed method is evaluated by solving various surface pdes such as surface laplace beltrami harmonic biharmonic triharmonic equations which are defined on the limit surfaces of extended loop subdivision for different initial control meshes numerical experiments show that the proposed method has desirable performance in terms of the accuracy convergence and computational cost for solving the above surface pdes defined on both open and closed surfaces the proposed approach is proved to be second order accuracy in the sense of l 2 norm with theoretical and or numerical results which is also outperformed over the standard linear finite element by several numerical comparisons

planar domain parameterization for isogeometric analysis based on teichmuller mapping abstract given the boundary curve of a planar domain finding a parametric spline representation for the domain is called domain parameterization a good parameterization of the computational domain plays a key role in isogeometric analysis since it influences the accuracy of the subsequent analysis in this paper we propose a new approach for planar domain parameterization based on teichmuller map a special map in the class of quasi conformal map under given correspondence of four boundary curves a unique teichmuller map between a unit square and a computational domain can be obtained which guarantees a bijection map and minimizes the maximal conformality distortion we propose an efficient iterative algorithm to compute the teichmuller map based on alternating direction method of multipliers admm experimental results show that our method can produce more uniform parameterization and increase the accuracy as well as decrease the condition numbers of the stiffness matrices in isogeometric analysis than other state of the art approaches

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

a new approach to solid modeling with trivariate t splines based on mesh optimization abstract we present a new method to construct a trivariate t spline representation of complex genus zero solids for the application of isogeometric analysis the proposed technique only demands a surface triangulation of the solid as input data the key of this method lies in obtaining a volumetric parameterization between the solid and the parametric domain the unitary cube to do that an adaptive tetrahedral mesh of the parametric domain is isomorphically transformed onto the solid by applying a mesh untangling and smoothing procedure the control points of the trivariate t spline are calculated by imposing the interpolation conditions on points sited both on the inner and on the surface of the solid the distribution of the interpolating points is adapted to the singularities of the domain in order to preserve the features of the surface triangulation

isogeometric analysis of the cahn hilliard phase field model abstract the cahn hilliard equation involves fourth order spatial derivatives finite element solutions are not common because primal variational formulations of fourth order operators are only well defined and integrable if the finite element basis functions are piecewise smooth and globally c 1 continuous there are a very limited number of two dimensional finite elements possessing c 1 continuity applicable to complex geometries but none in three dimensions we propose isogeometric analysis as a technology that possesses a unique combination of attributes for complex problems involving higher order differential operators namely higher order accuracy robustness two and three dimensional geometric flexibility compact support and most importantly the possibility of c 1 and higher order continuity a nurbs based variational formulation for the cahn hilliard equation was tested on two and three dimensional problems we present steady state solutions in two dimensions and for the first time in three dimensions to achieve these results an adaptive time stepping method is introduced we also present a technique for desensitizing calculations to dependence on mesh refinement this enables the calculation of topologically correct solutions on coarse meshes opening the way to practical engineering applications of phase field methodology

an improved algorithm for checking the injectivity of 2d toric surface patches abstract injective parametrizations are widely used both in theory and in applications the injectivity of parameteric curves and surfaces means that there are no self intersections toric surface patch is defined by a set of integer lattice points a and corresponding control points and weights which includes rational tensor product and triangle bezier patches as special cases in 2011 sottile and zhu presented a geometric method to check the injectivity of 2d toric surface patches in this paper we present an improved algorithm of their method the complexity of the improved algorithm is reduced from o n 3 to o n 2 where n a some examples are shown to illustrate the effectiveness of our algorithm moreover the algorithm is also applied to check the injectivity of parameterization in isogeometric analysis

studies of refinement and continuity in isogeometric structural analysis we investigate the effects of smoothness of basis functions on solution accuracy within the isogeometric analysis framework we consider two simple one dimensional structural eigenvalue problems and two static shell boundary value problems modeled with trivariate nurbs solids we also develop a local refinement strategy that we utilize in one of the shell analyses we find that increased smoothness that is the k method leads to a significant increase in accuracy for the problems of structural vibrations over the classical c0c0 continuous p method whereas a judicious insertion of c0c0 continuous surfaces about singularities in a mesh otherwise generated by the k method usually outperforms a mesh in which all basis functions attain their maximum level of smoothness we conclude that the potential for the k method is high but smoothness is an issue that is not well understood due to the historical dominance of c0c0 continuous finite elements and therefore further studies are warranted

efficient construction of multi block volumetric spline parameterization by discrete mask method isogeometric analysis attempts to unify the mathematical languages in design and analysis to realize the seamless integration of cad and cae in three dimensional isogeometric analysis parametric volume is employed as the computational domain for a given set of boundary information in this paper we propose a discrete mask method for the efficient construction of multi block volumetric parameterization based on a set of given boundary spline surfaces given the block partition information of a model the interior control points can be obtained efficiently by solving a sparse linear system the existence and uniqueness of the solution for the linear system are also proved after performing a pre process on the non compatible boundary surfaces the discrete coons mask is generalized to a unified form which provides more choices for the construction of inner control points the proposed method is not only suitable for the multi block case with c 1 continuity but can also be used for the case in which the boundary surfaces are of different degrees and with different number of control points and knot vectors several examples are presented to illustrate the effectiveness of the proposed method

an extension of klop s counterexample to the church rosser property to λ calculus with other ordered pair combinators abstract this note provides a short elementary proof of the fact that the reduction sequence discovered by klop 1979 provides a counterexample to the church rosser property for calculus or combinatory logic with ordered pair combinators it also extends the counterexample to systems with other forms of ordered pair combinators

parameterization of computational domain in isogeometric analysis methods and comparison parameterization of computational domain plays an important role in isogeometric analysis as mesh generation in finite element analysis in this paper we investigate this problem in the 2d case i e how to parametrize the computational domains by planar b spline surface from the given cad objects four boundary planar b spline curves firstly two kinds of sufficient conditions for injective b spline parameterization are derived with respect to the control points then we show how to find good parameterization of computational domain by solving a constraint optimization problem in which the constraint condition is the injectivity sufficient conditions of planar b spline parametrization and the optimization term is the minimization of quadratic energy functions related to the first and second derivatives of planar b spline parameterization by using this method the resulted parameterization has no self intersections and the isoparametric net has good uniformity and orthogonality after introducing a posteriori error estimation for isogeometric analysis we propose r refinement method to optimize the parameterization by repositioning the inner control points such that the estimated error is minimized several examples are tested on isogeometric heat conduction problem to show the effectiveness of the proposed methods and the impact of the parameterization on the quality of the approximation solution comparison examples with known exact solutions are also presented

algebraic grid generation on trimmed parametric surface using non self overlapping planar coons patch using a coons patch mapping to generate a structured grid in the parametric region of a trimmed surface can avoid the singularity of elliptic pde methods when only c1 continuous boundary is given the error of converting generic parametric c1 boundary curves into a specified representation form is also avoided however overlap may happen on some portions of the algebraically generated grid when a linear or naive cubic blending function is used in the mapping this severely limits its usage in most of engineering and scientific applications where a grid system of non self overlapping is strictly required to solve the problem non trivial blending functions in a coons patch mapping should be determined adaptively by the given boundary so that self overlapping can be averted we address the adaptive determination problem by a functional optimization method the governing equation of the optimization is derived by adding a virtual dimension in the parametric space of the given trimmed surface both one and two parameter blending functions are studied to resolve the difficulty of guessing good initial blending functions for the conjugate gradient method used a progressive optimization algorithm is then proposed which has been shown to be very effective in a variety of practical examples also an extension is added to the objective function to control the element shape finally experiment results are shown to illustrate the usefulness and effectiveness of the presented method copyright 2004 john wiley sons ltd

solid t spline construction from boundary representations for genus zero geometry abstract this paper describes a novel method to construct solid rational t splines for complex genus zero geometry from boundary surface triangulations we first build a parametric mapping between the triangulation and the boundary of the parametric domain a unit cube after that we adaptively subdivide the cube using an octree subdivision project the boundary nodes onto the input triangle mesh and at the same time relocate the interior nodes via mesh smoothing this process continues until the surface approximation error is less than a pre defined threshold t mesh is then obtained by pillowing the subdivision result one layer on the boundary and its quality is improved templates are implemented to handle extraordinary nodes and partial extraordinary nodes in order to get a gap free t mesh the obtained solid t spline is c 2 continuous except for the local region around each extraordinary node and partial extraordinary node the boundary surface of the solid t spline is c 2 continuous everywhere except for the local region around the eight nodes corresponding to the eight corners of the parametric cube finally a bezier extraction technique is used to facilitate t spline based isogeometric analysis the obtained bezier mesh is analysis suitable with no negative jacobians several examples are presented in this paper to show the robustness of the algorithm

adaptive isogeometric analysis by local h refinement with t splines isogeometric analysis based on non uniform rational b splines nurbs as basis functions preserves the exact geometry but suffers from the drawback of a rectangular grid of control points in the parameter space which renders a purely local refinement impossible this paper demonstrates how this difficulty can be overcome by using t splines instead t splines allow the introduction of so called t junctions which are related to hanging nodes in the standard fem obeying a few straightforward rules rectangular patches in the parameter space of the t splines can be subdivided and thus a local refinement becomes feasible while still preserving the exact geometry furthermore it is shown how state of the art a posteriori error estimation techniques can be combined with refinement by t splines numerical examples underline the potential of isogeometric analysis with t splines and give hints for further developments

elliptic grid generation based on laplace equations and algebraic transformations abstract an elliptic grid generation method is presented to generate boundary conforming grids in domains in 2d and 3d physical space and on minimal surfaces and parametrized surfaces in 3d physical space the elliptic grid generation method is based on the use of a composite mapping this composite mapping consists of a nonlinear transfinite algebraic transformation and an elliptic transformation the elliptic transformation is based on the laplace equations for domains or on the laplace beltrami equations for surfaces the algebraic transformation maps the computational space one to one onto a parameter space the elliptic transformation maps the parameter space one to one onto the domains or surfaces the composition of these two mappings is a differentiable one to one mapping from computational space onto the domains or surfaces and has a nonvanishing jacobian this composite mapping defines the grid point distribution in the interior of the domains or surfaces for domains and minimal surfaces the composite mapping obeys a nonlinear elliptic poisson system with control functions completely defined by the algebraic transformation the solution of the poisson systems is obtained by picard iteration and black box multigrid solvers for parametrized curved surfaces it is not necessary to define and solve a nonlinear elliptic poisson system instead a linear elliptic system and an inversion problem is solved to generate the grid in the interior of the surface

constructing analysis suitable parameterization of computational domain from cad boundary by variational harmonic method in isogeometric analysis parameterization of computational domain has great effects as mesh generation in finite element analysis in this paper based on the concept of harmonic mapping from the computational domain to parametric domain a variational harmonic approach is proposed to construct analysis suitable parameterization of computational domain from cad boundary for 2d and 3d isogeometric applications different from the previous elliptic mesh generation method in finite element analysis the proposed method focuses on isogeometric version and converts the elliptic pde into a nonlinear optimization problem in which a regular term is integrated into the optimization formulation to achieve more uniform and orthogonal iso parametric structure near convex concave parts of the boundary several examples are presented to show the efficiency of the proposed method in 2d and 3d isogeometric analysis