isogeometric boundary element analysis using unstructured t splines we couple collocated isogeometric boundary element methods and unstructured analysis suitable t spline surfaces for linear elastostatic problems we extend the definition of analysis suitable t splines to encompass unstructured control grids unstructured meshes and develop basis functions which are smooth rational polynomials defined in terms of the bezier extraction framework and which pass standard patch tests we then develop a collocation procedure which correctly accounts for sharp edges and corners extraordinary points and t junctions this approach is applied to several three dimensional problems including a real world t spline model of a propeller we believe this work clearly illustrates the power of combining new analysis suitable computer aided design technologies with established analysis methodologies in this case the boundary element method

geometrically consistent mesh modification a new paradigm of adaptivity is to execute refinement coarsening and smoothing of meshes on manifolds with incomplete information about their geometry and yet preserve position and curvature accuracy we refer to this collectively as geometrically consistent gc mesh modification we discuss the concept of discrete gc show the failure of naive approaches and propose and analyze a simple algorithm that is gc and accuracy preserving

isogeometric analysis of the advective cahn hilliard equation spinodal decomposition under shear flow we present a numerical study of the spinodal decomposition of a binary fluid undergoing shear flow using the advective cahn hilliard equation a stiff nonlinear parabolic equation characterized by the presence of fourth order spatial derivatives our numerical solution procedure is based on isogeometric analysis an approximation technique for which basis functions of high order continuity are employed these basis functions allow us to directly discretize the advective cahn hilliard equation without resorting to a mixed formulation we present steady state solutions for rectangular domains in two dimensions and for the first time in three dimensions we also present steady state solutions for the two dimensional taylor couette cell to enforce periodic boundary conditions in this curved domain we derive and utilize a new periodic bezier extraction operator we present an extensive numerical study showing the effects of shear rate surface tension and the geometry of the domain on the phase evolution of the binary fluid theoretical and experimental results are compared with our simulations

some estimates for h p k refinement in isogeometric analysis in this paper we propose a theoretical study of the approximation properties of nurbs spaces which are used in isogeometric analysis we obtain error estimates that are explicit in terms of the mesh size h the degree p and the global regularity measured by the parameter k our approach covers the approximation with global regularity from c 0 up to c k 1 with 2k 1 p notice that the interesting case of higher regularity up to k p is still open however our results give an indication of the role of the smoothness k in the approximation properties and offer a first mathematical justification of the potential of isogeometric analysis based on globally smooth nurbs

isogeometric analysis of geometric partial differential equations abstract we consider the numerical approximation of geometric partial differential equations pdes defined on surfaces in the 3d space in particular we focus on the geometric pdes deriving from the minimization of an energy functional by l 2 gradient flow we analyze two energy functionals the area which leads to the mean curvature flow a nonlinear second order pde and the willmore energy leading to the willmore flow a nonlinear fourth order pde we consider surfaces represented by single patch tensor product nurbs and discretize the pdes by means of nurbs based isogeometric analysis in the framework of the galerkin method to approximate the high order geometric pdes we use high order continuous nurbs basis functions for the time discretization of the nonlinear geometric pdes we use backward differentiation formulas bdf with extrapolation of the geometric quantities involved in the weak formulation of the problem in this manner we solve a linear problem at each time step we report numerical results concerning the mean curvature and willmore flows on different geometries of interest and we show the accuracy and efficiency of the proposed approximation scheme

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

on raghunathan s measure conjecture this paper represents the last in our three part series on raghunathan s measure conjecture see r1 r2 for parts i and ii more specifically let g be a real lie group all groups in this paper are assumed to be second countable with the lie algebra r a discrete subgroup of g and ir g r g the projection lwg rg the group g acts by right translations on r g x x g x e r g g e g let gt be a borel probability measure on r g define

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

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

a new finite element formulation for computational fluid dynamics x the compressible euler and navier stokes equations a space time element method is presented for solving the compressible euler and navier stokes equations the proposed formulation includes the variational equation predictor multi corrector algorithms and boundary conditions the variational equation is based on the time discontinuous galerkin method in which the physical entropy variables are employed a least squares operator and a discontinuity capturing operator are added resulting in a high order accurate and unconditionally stable method implicit explicit predictor multi corrector algorithms applicable to steady as well as unsteady problems are presented techniques are developed to enhance their efficiency implementation of boundary conditions is addressed in particular a technique is introduced to satisfy nonlinear essential boundary conditions and a consistent method is presented to calculate boundary fluxes numerical results are presented to demonstrate the performance of the method

invariant measures and the set of exceptions to littlewood s conjecture we classify the measures on sl k r sl k z which are invariant and ergodic under the action of the group a of positive diagonal matrices with positive entropy we apply this to prove that the set of exceptions to littlewood s conjecture has hausdorff dimension zero

computationally efficient solution to the cahn hilliard equation adaptive implicit time schemes mesh sensitivity analysis and the 3d isoperimetric problem we present an efficient numerical framework for analyzing spinodal decomposition described by the cahn hilliard equation we focus on the analysis of various implicit time schemes for two and three dimensional problems we demonstrate that significant computational gains can be obtained by applying embedded higher order runge kutta methods in a time adaptive setting this allows accessing time scales that vary by five orders of magnitude in addition we also formulate a set of test problems that isolate each of the sub processes involved in spinodal decomposition interface creation and bulky phase coarsening we analyze the error fluctuations using these test problems on the split form of the cahn hilliard equation solved using the finite element method with basis functions of different orders any scheme that ensures at least four elements per interface satisfactorily captures both sub processes our findings show that linear basis functions have superior error to cost properties this strategy coupled with a domain decomposition based parallel implementation let us notably augment the efficiency of a numerical cahn hillard solver and open new venues for its practical applications especially when three dimensional problems are considered we use this framework to address the isoperimetric problem of identifying local solutions in the periodic cube in three dimensions the framework is able to generate all five hypothesized candidates for the local solution of periodic isoperimetric problem in 3d sphere cylinder lamella doubly periodic surface with genus two lawson surface and triply periodic minimal surface p schwarz surface

variational problems and partial differential equations on implicit surfaces a novel framework for solving variational problems and partial differential equations for scalar and vector valued data defined on surfaces is introduced in this paper the key idea is to implicitly represent the surface as the level set of a higher dimensional function and to solve the surface equations in a fixed cartesian coordinate system using this new embedding function the equations are then both intrinsic to the surface and defined in the embedding space this approach thereby eliminates the need for performing complicated and inaccurate computations on triangulated surfaces as is commonly done in the literature we describe the framework and present examples in computer graphics and image processing applications including texture synthesis flow field visualization and image and vector field intrinsic regularization for data defined on 3d surfaces

a generalized α method for integrating the filtered navier stokes equations with a stabilized finite element method abstract a generalized method is developed and analyzed for linear first order systems the method is then extended to the filtered navier stokes equations within the context of a stabilized finite element method the formulation is studied through the application to laminar flow past a circular cylinder and turbulent flow past a long transverse groove the method is formulated to obtain a second order accurate family of time integrators whose high frequency amplification factor is the sole free parameter such an approach allows the replication of midpoint rule zero damping gear s method maximal damping or anything in between

an example of a non fourier mukai functor between derived categories of coherent sheaves orlov s famous representability theorem asserts that any fully faithful exact functor between the bounded derived categories of coherent sheaves on smooth projective varieties is a fourier mukai functor in this paper we show that this result is false without the fully faithfulness hypothesis we also show that our functor does not lift to the homotopy category of spectral categories if the ground field is mathbb q

afem for the laplace beltrami operator on graphs design and conditional contraction property we present an adaptive finite element method afem of any polynomial degree for the laplace beltrami operator on c 1 graphs in r d d 2 we first derive residual type a posteriori error estimates that account for the interaction of both the energy error in h 1 and the surface error in w 1 1 due to approximation of we devise a marking strategy to reduce the total error estimator namely a suitably scaled sum of the energy geometric and inconsistency error estimators we prove a conditional contraction property for the sum of the energy error and the total estimator the conditional statement encodes resolution of in w 1 1 we conclude with one numerical experiment that illustrates the theory

isogeometric analysis for second order partial differential equations on surfaces we consider the numerical solution of second order partial differential equations pdes on lower dimensional manifolds specifically on surfaces in three dimensional spaces for the spatial approximation we consider isogeometric analysis which facilitates the encapsulation of the exact geometrical description of the manifold in the analysis when this is represented by b splines or nurbs our analysis addresses linear nonlinear time dependent and eigenvalues problems involving the laplace beltrami operator on surfaces moreover we propose a priori error estimates under h refinement in the general case of second order pdes on the lower dimensional manifolds we highlight the accuracy and efficiency of isogeometric analysis with respect to the exactness of the geometrical representations of the surfaces

variational multiscale residual based turbulence modeling for large eddy simulation of incompressible flows we present an les type variational multiscale theory of turbulence our approach derives completely from the incompressible navier stokes equations and does not employ any ad hoc devices such as eddy viscosities we tested the formulation on forced homogeneous isotropic turbulence and turbulent channel flows in the calculations we employed linear quadratic and cubic nurbs a dispersion analysis of simple model problems revealed nurbs elements to be superior to classical finite elements in approximating advective and diffusive processes which play a significant role in turbulence computations the numerical results are very good and confirm the viability of the theoretical framework

scaled boundary parametrizations in isogeometric analysis abstract this paper deals with a special class of parametrizations for isogeometric analysis iga the so called scaled boundary parametrizations are easy to construct and particularly attractive if only a boundary description of the computational domain is available the idea goes back to the scaled boundary finite element method sb fem which has recently been extended to iga we take here a different viewpoint and study these parametrizations as bivariate or trivariate b spline functions that are directly suitable for standard galerkin based iga our main results are first a general framework for this class of parametrizations including aspects such as smoothness and regularity as well as a generalization to domains that are not star shaped second using the poisson equation as an example we explain the relation between standard galerkin based iga and the scaled boundary iga by means of the laplace beltrami operator and derive an equivalence theorem further results concern the separation of integrals in both approaches and an analysis of the singularity in the scaling center among the computational examples we present a rotor geometry that stems from a screw compressor machine and compare different parametrization strategies

anisotropic nurbs approximation in isogeometric analysis abstract we investigate the approximation error of mapped nurbs spaces such as those used in isogeometric analysis in accordance to the isoparametric paradigm we extend the results of 6 to the case of anisotropic meshes such as those needed in the presence of boundary layers the theoretical results cover anisotropic h refinement and are supported by numerical tests involving different geometries

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