block arnoldi based methods for large scale discrete time algebraic riccati equations in the present paper we present block arnoldi based methods for the computation of low rank approximate solutions of large discrete time algebraic riccati equations dare the proposed methods are projection methods onto block or extended block krylov subspaces we give new upper bounds for the norm of the error obtained by applying these block arnoldi based processes we also introduce the newton method combined with the block arnoldi algorithm and present some numerical experiments with comparisons between these methods

isogeometric analysis of the cahn hilliard equation a convergence study herein we present a numerical convergence study of the cahn hilliard phase field model within an isogeometric finite element analysis framework using a manufactured solution a mixed formulation of the cahn hilliard equation and the direct discretisation of the weak form which requires a c 1 continuous approximation are compared in terms of convergence rates for approximations that are higher than second order in space the direct discretisation is found to be superior suboptimal convergence rates occur when splines of order p 2 are used this is validated with a priori error estimates for linear problems the convergence analysis is completed with an investigation of the temporal discretisation second order accuracy is found for the generalised method this ensures the functionality of an adaptive time stepping scheme which is required for the efficient numerical solution of the cahn hilliard equation the isogeometric finite element framework is eventually validated by two numerical examples of spinodal decomposition

isogeometric numerical dispersion analysis for two dimensional elastic wave propagation abstract in this paper we carry out a numerical dispersion analysis for the linear two dimensional elastodynamics equations approximated by means of nurbs based isogeometric analysis in the framework of the galerkin method specifically we consider the analysis of harmonic plane waves in an isotropic and homogeneous elastic medium we compare and discuss the errors associated with the compressional and shear wave velocities and we provide the anisotropic curves for numerical approximations obtained by considering b spline and nurbs basis functions of different regularity namely globally c 0 and c p 1 continuous p being the polynomial degree we conclude our analysis by numerically simulating the seismic wave propagation in a sinusoidal shaped valley with discontinuous elastic parameters across an internal interface

smooth cubic spline spaces on unstructured quadrilateral meshes with particular emphasis on extraordinary points geometric design and isogeometric analysis considerations abstract we present a framework for geometric design and isogeometric analysis on unstructured quadrilateral meshes acknowledging the differing requirements posed by design e g the convenience of an intuitive control net and analysis e g good approximation behavior we propose the construction of a separate smooth spline space for each while ensuring isogeometric compatibility requiring the geometric models to be members of the analysis suitable spaces the methodology is simple and is presented for bi cubic splines extensions to higher degrees are possible and are briefly discussed the presentation has been structured to show compatibility with t splines a state of the art cad technology but the approach should extend to other locally refinable spline technologies based on local tensor product structures an instantiation of the framework is presented and several numerical tests focused on geometric design and isogeometric analysis demonstrate the versatility of the developed framework and show significantly higher convergence rates than attained previously in the considered setting

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

isogeometric analysis of high order partial differential equations on surfaces we consider the numerical approximation of high order partial differential equations pdes defined on surfaces in the three dimensional space with particular emphasis on closed surfaces we consider computational domains that can be represented by b splines or nurbs as for example the sphere and we spatially discretize the pdes by means of nurbs based isogeometric analysis in the framework of the standard galerkin method we numerically solve benchmark laplace beltrami problems of the fourth and sixth order as well as the corresponding eigenvalue problems with the goal of analyzing the role of the continuity of the nurbs basis functions on closed surfaces in this respect we show that the use of globally high order continuous basis functions as allowed by the construction of periodic nurbs leads to the efficient solution of the high order pdes finally we consider the numerical solution of high order phase field problems on closed surfaces namely the cahn hilliard and crystal equations

finite element approximation of the cahn hilliard equation with degenerate mobility we consider a fully practical finite element approximation of the cahn hilliard equation with degenerate mobility textstyle frac partial u partial t del b u del gamma lap u psi u where b cdot geq 0 is a diffusional mobility and psi cdot is a homogeneous free energy in addition to showing well posedness and stability bounds for our approximation we prove convergence in one space dimension furthermore an iterative scheme for solving the resulting nonlinear discrete system is analyzed we also discuss how our approximation has to be modified in order to be applicable to a logarithmic homogeneous free energy finally some numerical experiments are presented

triangulation based isogeometric analysis of the cahn hilliard phase field model abstract this paper presents triangulation based isogeometric analysis of the cahn hilliard phase field model the cahn hilliard phase field model is governed by a time dependent fourth order partial differential equation the corresponding primal variational form involves second order operators making it difficult to be directly analyzed with traditional c 0 finite element analysis in this paper we construct c 1 bernstein bezier simplicial elements through macro element techniques including various triangle split based macro elements in both 2d and 3d space we extend triangulation based isogeometric analysis to solving the primal variational form of the cahn hilliard equation we validate our method by convergence analysis showing the nodal and degree of freedom advantages over c 0 finite element analysis we then demonstrate detailed system evolution from randomly perturbed initial conditions in periodic two dimensional squares and three dimensional cubes we incorporate an adaptive time stepping scheme in these numerical experiments our numerical study demonstrates that triangulation based isogeometric analysis offers optimal convergence and time step stability is applicable to complex geometry and allows local refinement

solving the triharmonic equation over multi patch planar domains using isogeometric analysis abstract we present a framework for solving the triharmonic equation over bilinearly parameterized planar multi patch domains by means of isogeometric analysis our approach is based on the construction of a globally c 2 smooth isogeometric spline space which is used as discretization space the generated c 2 smooth space consists of three different types of isogeometric functions called patch edge and vertex functions all functions are entirely local with a small support and numerical examples indicate that they are well conditioned the construction of the functions is simple and works uniformly for all multi patch configurations while the patch and edge functions are given by a closed form representation the vertex functions are obtained by computing the null space of a small system of linear equations several examples demonstrate the potential of our approach for solving the triharmonic equation

a second order splitting method for the cahn hilliard equation a semi discrete finite element method requiring only continuous element is presented for the approximation of the solution of the evolutionary fourth order in space cahn hilliard equation optimal order error bounds are derived in various norms for an implementation which uses mass lumping the continuous problem has an energy based lyapunov functional it is proved that this property holds for the discrete problem

space of c2 smooth geometrically continuous isogeometric functions on planar multi patch geometries dimension and numerical experiments abstract we study the space of c 2 smooth isogeometric functions on bilinearly parameterized multi patch domains r 2 where the graph of each isogeometric function is a multi patch spline surface of bidegree d d d 5 6 the space is fully characterized by the equivalence of the c 2 smoothness of an isogeometric function and the g 2 smoothness of its graph surface cf groisser and peters 2015 kapl et al 2015 this is the reason to call its functions c 2 smooth geometrically continuous isogeometric functions in particular the dimension of this c 2 smooth isogeometric space is investigated the study is based on the decomposition of the space into three subspaces and is an extension of the work kapl and vitrih 2017 to the multi patch case in addition we present an algorithm for the construction of a basis and use the resulting globally c 2 smooth functions for numerical experiments such as performing l 2 approximation and solving triharmonic equation on bilinear multi patch domains the numerical results indicate optimal approximation order

comparative computational analysis of the cahn hilliard equation with emphasis on c1 continuous methods the numerical treatment of the fourth order cahn hilliard equation is nonstandard using a galerkin method necessitates for instance piecewise smooth and globally c 1 continuous basis functions or a mixed formulation the latter is obtained introducing an auxiliary field which allows to rephrase the cahn hilliard equation as a set of two coupled second order equations in view of this the formulation in terms of the primal unknown appears to be a more intuitive and natural choice but requires a c 1 continuous interpolation therefore isogeometric analysis using a spline basis and natural element analysis are addressed in the present contribution mixed second order finite element methods introducing the chemical potential or alternatively a nonlocal concentration as auxiliary field serve as references to which both higher order methods are compared in terms of accuracy and efficiency

isogeometric analysis of the navier stokes cahn hilliard equations with application to incompressible two phase flows abstract in this work we provide a unified and comparative description of the most prominent phase field based two phase flow models and present our numerical results of the application of galerkin based isogeometric analysis iga to incompressible navier stokes cahn hilliard nsch equations in velocity pressure phase field chemical potential formulation for the approximation of the velocity and pressure fields lbb compatible non uniform rational b spline spaces are used which can be regarded as smooth generalizations of taylor hood pairs of finite element spaces the one step scheme is used for the discretization in time the static and rising bubble in addition to the nonlinear rayleigh taylor instability flow problems are considered in two dimensions as model problems in order to investigate the numerical properties of the scheme

isogeometric analysis of the isothermal navier stokes korteweg equations this paper is devoted to the numerical simulation of the navier stokes korteweg equations a phase eld model for water water vapor two phase ows we develop a numerical formulation based on isogeometric analysis that permits straightforward treatment of the higher order partial dierenti al operator that represents capillarity we introduce a new renement methodology that desensitizes the numerical solution to the computational mesh and achieves mesh invariant solutions finally we present several numerical examples in two and three dimensions that illustrate the eectiveness and robustness of our approach

isogeometric approximation of cardiac electrophysiology models on surfaces an accuracy study with application to the human left atrium abstract we consider isogeometric analysis in the framework of the galerkin method for the spatial approximation of cardiac electrophysiology models defined on nurbs surfaces specifically we perform a numerical comparison between basis functions of degree p 1 and globally c k continuous with k 0 or p 1 to find the most accurate approximation of a propagating front with the minimal number of degrees of freedom we show that b spline basis functions of degree p 1 which are c p 1 continuous capture accurately the front velocity of the transmembrane potential even with moderately refined meshes similarly we show that for accurate tracking of curved fronts high order continuous b spline basis functions should be used finally we apply isogeometric analysis to an idealized human left atrial geometry described by nurbs with physiologically sound fiber directions and anisotropic conductivity tensor to demonstrate that the numerical scheme retains its favorable approximation properties also in a more realistic setting

robustness of isogeometric structural discretizations under severe mesh distortion abstract this paper investigates higher order and higher continuity functions in isogeometric structural analysis under distortion of the control and physical meshes first the concepts behind isogeometric analysis are briefly reviewed the effect of perturbing control points upon the geometrical mapping is observed in the one dimensional setting the concept is extended to multiple dimensions with a series of cases where the interior of a cube is distorted in different ways and patch tests are performed to assess accuracy a circular fillet patch is introduced through the appropriate degeneration of a control mesh in order to address the problem of modeling small features in finite element models the fillet patch is tested with two problems of linear elasticity and is shown to be accurate the investigation is further extended to the setting of large deformation nearly incompressible nonlinear elasticity with two examples which make use of the recently proposed f projection method

computationally efficient approach for the minimization of volume constrained vector valued ginzburg landau energy functional the minimization of volume constrained vector valued ginzburg landau energy functional is considered in the present study it has many applications in computational science and engineering like the conservative phase separation in multiphase systems such as the spinodal decomposition phase coarsening in multiphase systems color image segmentation and optimal space partitioning a computationally efficient algorithm is presented to solve the space discretized form of the original optimization problem the algorithm is based on the constrained nonmonotone l 2 gradient flow of ginzburg landau functional followed by a regularization step which is resulted from the tikhonov regularization term added to the objective functional that lifts the solution from the l 2 function space into h 1 space the regularization step not only improves the convergence rate of the presented algorithm but also increases its stability bound the step size selection based on the barzilai borwein approach is adapted to improve the convergence rate of the introduced algorithm the success and performance of the presented approach is demonstrated throughout several numerical experiments to make it possible to reproduce the results presented in this work the matlab implementation of the presented algorithm is provided as the supplementary material developing an efficient algorithm to solve conservative gradient flow of multiphase ginzburg landau functional performing the constrained minimization of energy functional on the h 1 space using non monotone optimization path by applying the barzilai borwein step size selection approach the matlab implementation of the presented algorithms is given as supplementary material

liquid vapor phase transition thermomechanical theory entropy stable numerical formulation and boiling simulations abstract we develop a new continuum mechanics modeling framework for liquid vapor flows with particular focus on the van der waals fluid by invoking microforce theory the coleman noll procedure is generalized to derive consistent constitutive relations in the presence of non local effects a new thermodynamically consistent algorithm for the van der waals model is designed using functional entropy variables and a new temporal scheme employing a family of new quadrature rules we show that the resulting fully discrete scheme is unconditionally stable in entropy and second order time accurate isogeometric analysis is utilized for spatial discretization the analytical properties of the formulation are corroborated by benchmark problems three sets of application problems are simulated to demonstrate the capability of the model and the algorithm our methodology provides a particularly useful predictive tool for boiling flows

variational boundary conditions based on the nitsche method for fitted and unfitted isogeometric discretizations of the mechanically coupled cahn hilliard equation the primal variational formulation of the fourth order cahn hilliard equation requires c1 continuous finite element discretizations e g in the context of isogeometric analysis in this paper we explore the variational imposition of essential boundary conditions that arise from the thermodynamic derivation of the cahn hilliard equation in primal variables our formulation is based on the symmetric variant of nitsche s method does not introduce additional degrees of freedom and is shown to be variationally consistent in contrast to strong enforcement the new boundary condition formulation can be naturally applied to any mapped isogeometric parametrization of any polynomial degree in addition it preserves full accuracy including higher order rates of convergence which we illustrate for boundary fitted discretizations of several benchmark tests in one two and three dimensions unfitted cartesian b spline meshes constitute an effective alternative to boundary fitted isogeometric parametrizations for constructing c1 continuous discretizations in particular for complex geometries we combine our variational boundary condition formulation with unfitted cartesian b spline meshes and the finite cell method to simulate chemical phase segregation in a composite electrode this example involving coupling of chemical fields with mechanical stresses on complex domains and coupling of different materials across complex interfaces demonstrates the flexibility of variational boundary conditions in the context of higher order unfitted isogeometric discretizations

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

dirichlet forms and associated heat kernels on the cantor set induced by random walks on trees abstract transient random walk on a tree induces a dirichlet form on its martin boundary which is the cantor set the procedure of the inducement is analogous to that of the douglas integral on s 1 associated with the brownian motion on the unit disk in this paper those dirichlet forms on the cantor set induced by random walks on trees are investigated explicit expressions of the hitting distribution harmonic measure and the induced dirichlet form on the cantor set are given in terms of the effective resistances an intrinsic metric on the cantor set associated with the random walk is constructed under the volume doubling property of with respect to the intrinsic metric asymptotic behaviors of the heat kernel the jump kernel and moments of displacements of the process associated with the induced dirichlet form are obtained furthermore relation to the noncommutative riemannian geometry is discussed

computation of multiphase systems with phase field models phase field models offer a systematic physical approach for investigating complex multiphase systems behaviors such as near critical interfacial phenomena phase separation under shear and microstructure evolution during solidification however because interfaces are replaced by thin transition regions diffuse interfaces phase field simulations require resolution of very thin layers to capture the physics of the problems studied this demands robust numerical methods that can efficiently achieve high resolution and accuracy especially in three dimensions we present here an accurate and efficient numerical method to solve the coupled cahn hilliard navier stokes system known as model h that constitutes a phase field model for density matched binary fluids with variable mobility and viscosity the numerical method is a time split scheme that combines a novel semi implicit discretization for the convective cahn hilliard equation with an innovative application of high resolution schemes employed for direct numerical simulations of turbulence this new semi implicit discretization is simple but effective since it removes the stability constraint due to the nonlinearity of the cahn hilliard equation at the same cost as that of an explicit scheme it is derived from a discretization used for diffusive problems that we further enhance to efficiently solve flow problems with variable mobility and viscosity moreover we solve the navier stokes equations with a robust time discretization of the projection method that guarantees better stability properties than those for crank nicolson based projection methods for channel geometries the method uses a spectral discretization in the streamwise and spanwise directions and a combination of spectral and high order compact finite difference discretizations in the wall normal direction the capabilities of the method are demonstrated with several examples including phase separation with and without shear in two and three dimensions the method effectively resolves interfacial layers of as few as three mesh points the numerical examples show agreement with analytical solutions and scaling laws where available and the 3d simulations in the presence of shear reveal rich and complex structures including strings

quasi incompressible cahn hilliard fluids and topological transitions one of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect it is well known that classical methods involving sharp interfaces fail to describe this type of phenomena following some previous work in this area we suggest a physically motivated regularization of the euler equations which allows topological transitions to occur smoothly in this model the sharp interface is replaced by a narrow transition layer across which the fluids may mix the model describes a flow of a binary mixture and the internal structure of the interface is determined by both diffusion and motion an advantage of our regularization is that it automatically yields a continuous description of surface tension which can play an important role in topological transitions an additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the euler or navier stokes and the cahn hilliard equations the model takes into account weak non locality dispersion associated with an internal length scale and localized dissipation due to mixing the non locality introduces a dimensional surface energy dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes in particular we study a non trivial limit when both components are incompressible the pressure is kinematic but the velocity field is non solenoidal quasi incompressibility to demonstrate the effects of quasi incompressibility we analyse the linear stage of spinodal decomposition in one dimension we show that when the densities of the fluids are not perfectly matched the evolution of the concentration field causes fluid motion even if the fluids are inviscid in the limit of infinitely thin and well separated interfacial layers an appropriately scaled quasi incompressible euler cahn hilliard system converges to the classical sharp interface model in order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity

multi component cahn hilliard system with different boundary conditions in complex domains abstract we propose an efficient phase field model for multi component cahn hilliard ch systems in complex domains the original multi component cahn hilliard system with a fixed phase is modified in order to make it suitable for complex domains in the cartesian grid along with contact angle or no mass flow boundary conditions on the complex boundaries the proposed method uses a practically unconditionally gradient stable nonlinear splitting numerical scheme further a nonlinear full approximation storage multigrid algorithm is used for solving semi implicit formulations of the multi component ch system incorporated with an adaptive mesh refinement technique the robustness of the proposed method is validated through various numerical simulations including multi phase separations via spinodal decomposition equilibrium contact angle problems and multi phase flows with a background velocity field in complex domains

parametric mesh regularization for interpolatory shape design and isogeometric analysis over a mesh of arbitrary topology abstract isogeometric analysis iga over a mesh of arbitrary topology is a challenging task since the concept of iga was proposed in the literature one can find some results in iga over an irregular mesh based on approximating schemes while little work is reported so far using interpolatory schemes this paper presents a parametric mesh regularization pmr method for iga using mapped interpolatory basis functions over a mesh of arbitrary topology given a geometric model defined over a mesh of arbitrary topology a global parametric mesh is first defined from the given control mesh using a gravity center method gcm yuan and ma 2014 the resulting parametric mesh is further regularized such that it ensures absolute interpolatory modeling with mapped interpolatory basis functions necessary geometric rules are developed for the regularization process while maintaining the same mesh topological structure the interpolatory basis function used in this paper is a kind of truncated interpolatory basic function tibf studied in our early work yuan and ma 2013 it can also be applied to other mapped interpolatory basis functions in addition to the capability of interpolatory modeling the proposed pmr method can also be applied to approximating schemes which further enriches the capability of such schemes in both modeling and isogeometric analysis of some models in a more appropriate and convenient manner the proposed pmr method has also the potential in achieving higher order continuity at extraordinary vertices using higher order mapped interpolatory basis functions

isogeometric finite element data structures based on bezier extraction of t splines we develop finite element data structures for t splines based on bezier extraction generalizing our previous work for nurbs as in traditional finite element analysis the extracted bezier elements are defined in terms of a fixed set of polynomial basis functions the so called bernstein basis the bezier elements may be processed in the same way as in a standard finite element computer program utilizing exactly the same data processing arrays in fact only the shape function subroutine needs to be modified while all other aspects of a finite element program remain the same a byproduct of the extraction process is the element extraction operator this operator localizes the topological and global smoothness information to the element level and represents a canonical treatment of t junctions referred to as hanging nodes in finite element analysis and a fundamental feature of t splines a detailed example is presented to illustrate the ideas copyright 2011 john wiley sons ltd

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

provably unconditionally stable second order time accurate mixed variational methods for phase field models we introduce provably unconditionally stable mixed variational methods for phase field models our formulation is based on a mixed finite element method for space discretization and a new second order accurate time integration algorithm the fully discrete formulation inherits the main characteristics of conserved phase dynamics namely mass conservation and nonlinear stability with respect to the free energy we illustrate the theory with the cahn hilliard equation but our method may be applied to other phase field models we also propose an adaptive time stepping version of the new time integration method we present some numerical examples that show the accuracy stability and robustness of the new method

on the convergence of inexact newton methods for discrete time algebraic riccati equations abstract in this paper we present a convergence analysis of the inexact newton method for solving discrete time algebraic riccati equations dares for large and sparse systems the inexact newton method requires at each iteration the solution of a symmetric stein matrix equation these linear matrix equations are solved approximatively by the alternating directions implicit adi or smith s methods we give some new matrix identities that will allow us to derive new theoretical convergence results for the obtained inexact newton sequences we show that under some necessary conditions the approximate solutions satisfy some desired properties such as the d stability the theoretical results developed in this paper are an extension to the discrete case of the analysis performed by feitzinger et al 2009 8 for the continuous time algebraic riccati equations in the last section we give some numerical experiments

diffuse interface model for incompressible two phase flows with large density ratios we investigate the applicability of an incompressible diffuse interface model for two phase incompressible fluid flows with large viscosity and density contrasts diffuse interface models have been used previously primarily for density matched fluids and it remains unclear to what extent such models can be used for fluids of different density thereby potentially limiting the application of these models in this paper the convective cahn hilliard equation and the condition that the velocity field is divergence free are derived from the conservation law of mass of binary mixtures in a straightforward way for fluids with large density and viscosity ratios differences in the equations of motion with a previously derived quasi incompressible model are shown to result from the respective assumptions made regarding the relationship between the diffuse fluxes of two species the convergence properties of the model are investigated for cases with large density ratio quantitative comparisons are made with results from previous studies to validate the model and its numerical implementation tests show that the variation in volume during the computation is of the order of machine accuracy which is consistent with our use of a conservative discretization scheme finite volume methods for the cahn hilliard equation results of the method are compared with previous work for the change in topology of rising bubbles and rayleigh taylor instability additional results are presented for head on droplet collision and the onset of droplet entrainment in stratified flows

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

dispersion optimized quadrature rules for isogeometric analysis modified inner products their dispersion properties and optimally blended schemes abstract this paper introduces optimally blended quadrature rules for isogeometric analysis and analyzes the numerical dispersion of the resulting discretizations to quantify the approximation errors when we modify the inner products we generalize the pythagorean eigenvalue theorem of strang and fix the proposed blended quadrature rules have advantages over alternative integration rules for isogeometric analysis on uniform and non uniform meshes as well as for different polynomial orders and continuity of the basis the optimally blended schemes improve the convergence rate of the method by two orders with respect to the fully integrated galerkin method the proposed technique increases the accuracy and robustness of isogeometric analysis for wave propagation problems

weak coupling for isogeometric analysis of non matching and trimmed multi patch geometries nitsche s method can be used as a coupling tool for non matching discretizations by weakly enforcing interface constraints we explore the use of weak coupling based on nitsche s method in the context of higher order and higher continuity b splines and nurbs we demonstrate that weakly coupled spline discretizations do not compromise the accuracy of isogeometric analysis we show that the combination of weak coupling with the finite cell method opens the door for a truly isogeometric treatment of trimmed b spline and nurbs geometries that eliminates the need for costly reparameterization procedures we test our methodology for several relevant technical problems in two and three dimensions such as gluing together trimmed multi patches and connecting non matching meshes that contain b spline basis functions and standard triangular finite elements the results demonstrate that the concept of nitsche based weak coupling in conjunction with the finite cell method has the potential to considerably increase the flexibility of the design through analysis process in isogeometric analysis

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

a class of stable spectral methods for the cahn hilliard equation in this work the initial boundary value problem of two dimensional cahn hilliard equation is considered a class of fully discrete dissipative fourier spectral schemes are proposed moreover semi implicit prediction correction schemes are presented the numerical simulations are performed to demonstrate the effectiveness of the proposed schemes

isogeometric analysis for phase field models of geometric pdes and high order pdes on stationary and evolving surfaces abstract in this paper we present an isogeometric analysis iga for phase field models of three different yet closely related classes of partial differential equations pdes geometric pdes high order pdes on stationary surfaces and high order pdes on evolving surfaces the latter can be a coupling of the former two classes in the context of geometric pdes we consider mean curvature flow and willmore flow problems and their corresponding phase field approximations which yield second order and fourth order nonlinear parabolic pdes through some numerical examples we study the convergence behavior of isogeometric analysis for these equations using the method of manufactured solutions moreover we study numerically the convergence of these phase field approximations to the sharp interface solutions as for the high order pdes on stationary surfaces we consider a model problem which is the cahn hilliard equation on a unit sphere where the surface is modeled using a diffuse interface approach finally as a model problem for high order pdes on evolving surfaces we consider a phase field model of a deforming multicomponent vesicle which couples the vesicle shape changes with the phase separation process on the vesicle surface the model consists of two fourth order nonlinear pdes which their direct finite element formulation in a galerkin framework necessitates smooth basis functions with at least global c 1 continuity a condition that can be easily satisfied using spline bases in iga we solve the coupled equations both in two dimensions where the vesicle is a curve and in three dimensions where the vesicle is a surface the simulation results agree with the numerical and experimental results from the literature

c1 macroelements in adaptive finite element methods some interesting properties of composite basis functions for c1 macroelements are investigated including their use for constructing conforming c1 function spaces on non conforming adaptively refined meshes of particular interest is the classical cubic hsieh clough tocher 3 split triangle because of its simplicity and convergence properties in fourth order problems as compared with the powell sabin heindl 6 split and 12 split triangles a posteriori error indicators for adaptive refinement are developed numerical experiments demonstrate convergence rates and adaptive refinement performance based on a simplified error indicator is tested extensibility to analogous three dimensional tetrahedral elements is briefly discussed copyright 2006 john wiley sons ltd

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 interactive geometry modeling and parametric design platform for isogeometric analysis in this paper an interactive parametric design through analysis platform is proposed to help design engineers and analysts make more effective use of isogeometric analysis iga to improve their product design and performance we develop several rhinoceros rhino plug ins to take input design parameters through a user friendly interface generate appropriate surface and or volumetric models perform mechanical analysis and visualize the solution fields all within the same computer aided design cad program as part of this effort we propose and implement graphical generative algorithms for iga model creation and visualization based on grasshopper a visual programming interface to rhino the developed platform is demonstrated on two structural mechanics examples an actual wind turbine blade and a model of an integrally bladed rotor ibr in the latter example we demonstrate how the rhino functionality may be utilized to create conforming volumetric models for iga

a fractional phase field model for two phase flows with tunable sharpness algorithms and simulations abstract we develop a fractional extension of a mass conserving allen cahn phase field model that describes the mixture of two incompressible fluids the fractional order controls the sharpness of the interface which is typically diffusive in integer order phase field models the model is derived based on an energy variational formulation an additional constraint is employed to make the allen cahn formulation mass conserving and comparable to the cahn hilliard formulation but at reduced cost the spatial discretization is based on a petrov galerkin spectral method whereas the temporal discretization is based on a stabilized adi scheme both for the phase field equation and for the navier stokes equation we demonstrate the spectral accuracy of the method with fabricated smooth solutions and also the ability to control the interface thickness between two fluids with different viscosity and density in simulations of two phase flow in a pipe and of a rising bubble we also demonstrate that using a formulation with variable fractional order we can deal simultaneously with both erroneous boundary effects and sharpening of the interface at no extra resolution

some remarks on spectral convergence and stability of iso geometric analysis abstract we study convergence and stability of iso geometric analysis or b spline method under k refinement for splines of maximum smoothness in the setting of the initial value problem for the heat equation spectral error bounds for the semi discrete approximation are shown for the periodic and dirichlet problems on the unit cube in r d time stepping restrictions for the stability of the scheme are bounded in the case 1 2 1 with the help of the best constants in the inverse estimates for splines in particular in the periodic case the time stepping does not become more restrictive as k increases in addition we show how the fast fourier transform can be used for an efficient implementation of the scheme for problems with periodic boundary conditions

isogeometric analysis with c1 hierarchical functions on planar two patch geometries abstract adaptive isogeometric methods for the solution of partial differential equations rely on the construction of locally refinable spline spaces a simple and efficient way to obtain these spaces is to apply the multi level construction of hierarchical splines that can be used on single patch domains or in multi patch domains with c 0 continuity across the patch interfaces due to the benefits of higher continuity in isogeometric methods recent works investigated the construction of spline spaces with global c 1 continuity on two or more patches in this paper we show how these approaches can be combined with the hierarchical construction to obtain global c 1 continuous hierarchical splines on two patch domains a selection of numerical examples is presented to highlight the features and effectivity of the construction

multi degree smooth polar splines a framework for geometric modeling and isogeometric analysis we develop a multi degree polar spline framework with applications to both geometric modeling and isogeometric analysis first multi degree splines are introduced as piecewise non uniform rational b splines nurbs of non uniform or variable polynomial degree and a simple algorithm for their construction is presented then an extension to two dimensional polar configurations is provided by means of a tensor product construction with a collapsed edge suitable combinations of these basis functions encoded in a so called isogeometric analysis suitable extraction operator yield ckck smooth polar splines for any k 0k 0 we show that it is always possible to construct a set of smooth polar spline basis functions that form a convex partition of unity and possess locality explicit constructions for k 0 1 2 k 0 1 2 are presented optimal approximation behavior is observed numerically and examples of applications to free form design smooth hole filling and high order partial differential equations demonstrate the applicability of the developed framework

unconditionally energy stable time stepping scheme for cahn morral equation an unconditionally energy stable time stepping scheme is introduced to solve cahn morral like equations in the present study it is constructed based on the combination of david eyre s time stepping scheme and schur complement approach although the presented method is general and independent of the choice of homogeneous free energy density function term logarithmic and polynomial energy functions are specifically considered in this paper the method is applied to study the spinodal decomposition in multi component systems and optimal space tiling problems a penalization strategy is developed in the case of later problem to avoid trivial solutions extensive numerical experiments demonstrate the success and performance of the presented method according to the numerical results the method is convergent and energy stable independent of the choice of time stepsize its matlab implementation is included in the appendix for the numerical evaluation of algorithm and reproduction of the presented results extension of eyre s convex concave splitting scheme to multiphase systems efficient solution of spinodal decomposition in multi component systems efficient solution of least perimeter periodic space partitioning problem developing a penalization strategy to avoid trivial solutions presentation of matlab implementation of the introduced algorithm

continuous discontinuous finite element approximations of fourth order elliptic problems in structural and continuum mechanics with applications to thin beams and plates and strain gradient elasticity abstract a new finite element method for fourth order elliptic partial differential equations is presented and applied to thin bending theory problems in structural mechanics and to a strain gradient theory problem the method combines concepts from the continuous galerkin cg method the discontinuous galerkin dg method and stabilization techniques a brief review of the cg method the dg method and stabilization techniques highlights the advantages and disadvantages of these methods and suggests a new approach for the solution of fourth order elliptic problems a continuous discontinuous galerkin c dg method is proposed which uses c 0 continuous interpolation functions and is formulated in the primary variable only the advantage of this formulation over a more traditional mixed approach is that the introduction of additional unknowns and related difficulties can be avoided in the context of thin bending theory the c dg method leads to a formulation where displacements are the only degrees of freedom and no rotational degrees of freedom need to be considered the main feature of the c dg method is the weak enforcement of continuity of first and higher order derivatives through stabilizing terms on interior boundaries consistency stability and convergence of the method are shown analytically numerical experiments verify the theoretical results and applications are presented for bernoulli euler beam bending poisson kirchhoff plate bending and a shear layer problem using toupin mindlin strain gradient theory

an energy stable time integrator for phase field models we introduce a provably energy stable time integration method for general classes of phase field models with polynomial potentials we demonstrate how taylor series expansions of the nonlinear terms present in the partial differential equations of these models can lead to expressions that guarantee energy stability implicitly which are second order accurate in time the spatial discretization relies on a mixed finite element formulation and isogeometric analysis we also propose an adaptive time stepping discretization that relies on a first order backward approximation to give an error estimator this error estimator is accurate robust and does not require the computation of extra solutions to estimate the error this methodology can be applied to any second order accurate time integration scheme we present numerical examples in two and three spatial dimensions which confirm the stability and robustness of the method the implementation of the numerical schemes is done in petiga a high performance isogeometric analysis framework

functional entropy variables a new methodology for deriving thermodynamically consistent algorithms for complex fluids with particular reference to the isothermal navier stokes korteweg equations we propose a new methodology for the numerical solution of the isothermal navier stokes korteweg equations our methodology is based on a semi discrete galerkin method invoking functional entropy variables a generalization of classical entropy variables and a new time integration scheme we show that the resulting fully discrete scheme is unconditionally stable in energy second order time accurate and mass conservative we utilize isogeometric analysis for spatial discretization and verify the aforementioned properties by adopting the method of manufactured solutions and comparing coarse mesh solutions with overkill solutions various problems are simulated to show the capability of the method our methodology provides a means of constructing unconditionally stable numerical schemes for nonlinear non convex hyperbolic systems of conservation laws

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

regular article calculation of two phase navier stokes flows using phase field modeling phase field models provide a way to model fluid interfaces as having finite thickness this can allow the computation of interface movement and deformation on fixed grids this paper applies phase field modeling to the computation of two phase incompressible navier stokes flows the navier stokes equations are modified by the addition of the continuum forcing email protected f where c is the composition variable and f is c s chemical potential the equation for interface advection is replaced by a continuum advective diffusion equation with diffusion driven by c s chemical potential gradients the paper discusses how solutions to these equations approach those of the original sharp interface navier stokes equations as the interface thickness e and the diffusivity both go to zero the basic flow physics of phase field interfaces is discussed straining flows can thin or thicken an interface and this must be resisted by a high enough diffusion on the other hand too large a diffusion will overly damp the flow these two constraints result in an upper bound for the diffusivity of o e and a lower bound of o e 2 within these two bounds the phase field navier stokes equations appear to generate an o e error relative to the exact sharp interface equations an o h 2 e 2 numerical method is introduced that is energy conserving in the sense that creation of interface energy by convection is always balanced by an equal decrease in kinetic energy caused by surface tension forcing an o h 4 e 4 compact scheme is introduced that takes advantage of the asymptotic comparatively smooth behavior of the chemical potential for o e accurate phase field models the optimum path to convergence for this scheme appears to be email protected 4 5 the asymptotic rate of convergence corresponding to this is o h 4 5 but results at practical resolutions show that the practical convergence of the method is generally considerably faster than linear extensive analysis and computations show that this scheme is very effective and accurate it allows the accurate calculation of two phase flows with interfaces only two cells wide computational results are given for linear capillary waves and for rayleigh taylor instabilities the first set of computations is compared to exact solutions of the diffuse interface equations and of the original sharp interface equations the rayleigh taylor computations test the ability of the method to compute highly deforming flows these flows include near singular phenomena such as interface coalescences and breakups contact line movement and the formation and breakup of thin wall films grid refinement studies are made and rapid convergence is found for macroscopic flow features such as instability growth rate and propagation speed wavelength and the general physical characteristics of the instability and mass transfer rates

isogeometric collocation cost comparison with galerkin methods and extension to adaptive hierarchical nurbs discretizations abstract we compare isogeometric collocation with isogeometric galerkin and standard c 0 finite element methods with respect to the cost of forming the matrix and residual vector the cost of direct and iterative solvers the accuracy versus degrees of freedom and the accuracy versus computing time on this basis we show that isogeometric collocation has the potential to increase the computational efficiency of isogeometric analysis and to outperform both isogeometric galerkin and standard c 0 finite element methods when a specified level of accuracy is to be achieved with minimum computational cost we then explore an adaptive isogeometric collocation method that is based on local hierarchical refinement of nurbs basis functions and collocation points derived from the corresponding multi level greville abscissae we introduce the concept of weighted collocation that can be consistently developed from the weighted residual form and the two scale relation of b splines using weighted collocation in the transition regions between hierarchical levels we are able to reliably handle coincident collocation points that naturally occur for multi level greville abscissae the resulting method combines the favorable properties of isogeometric collocation and hierarchical refinement in terms of computational efficiency local adaptivity robustness and straightforward implementation which we illustrate by numerical examples in one two and three dimensions

on the use of local maximum entropy approximants for cahn hilliard phase field models in 2d domains and on surfaces abstract we apply a local maximum entropy lme approximation scheme to the fourth order phase field model for the traditional cahn hilliard theory the discretization of the cahn hilliard equation by galerkin method requires at least c 1 continuous basis functions this requirement can be fulfilled by using lme shape functions which are c continuous in this case the primal variational formulations of the fourth order partial differential equation are well defined and integrable hence there is no need to split the fourth order partial differential equation into two second order partial differential equations this splitting scheme is a common practice in mixed finite element formulations with c 0 continuous lagrange shape functions furthermore we use a general and simple numerical method such as statistical manifold learning technique that allows dealing with general point set surfaces avoiding a global parameterization which can be applied to tackle surfaces of complex geometry and topology and to solve cahn hilliard equation on general surfaces finally the flexibility and robustness of the presented methodology is demonstrated for several representative numerical examples

an isogeometric analysis approach to gradient dependent plasticity summary gradient dependent plasticity can be used to achieve mesh objective results upon loss of well posedness of the initial boundary value problem due to the introduction of strain softening non associated flow and geometric non linearity a prominent class of gradient plasticity models considers a dependence of the yield strength on the laplacian of the hardening parameter usually an invariant of the plastic strain tensor this inclusion causes the consistency condition to become a partial differential equation in addition to the momentum balance at the internal moving boundary one has to impose appropriate boundary conditions on the hardening parameter or equivalently on the plastic multiplier this internal boundary condition can be enforced without tracking the elastic plastic boundary by requiring c1 continuity with respect to the plastic multiplier in this contribution this continuity has been achieved by using nurbs as shape functions both for the plastic multiplier and for the displacements one advantage of this isogeometric analysis approach is that the displacements can be interpolated one order higher making it consistent with the interpolation of the plastic multiplier this is different from previous approaches which have been exploited the regularising effect of gradient plasticity is shown for one and two dimensional boundary value problems this article is protected by copyright all rights reserved

rectified unstructured t splines with dynamic weighted refinement for improvement in geometric consistency and approximation convergence abstract t splines developed on unstructured meshes are called unstructured t splines which have the problem of geometric inconsistency and unfairness of mesh density evolution over regular or irregular elements during the refinement geometric inconsistency means that the geometry defined by an unstructured t spline keeps changing throughout the refinement the unfairness of mesh density evolution pertains to different scalings of regular and irregular elements during the refinement which drags down the convergence of the approximation result in isogeometric analysis this paper proposes rectified unstructured t splines to solve these two problems with the dynamic weighted refinement and rectified t spline basis functions owing to the dynamic weighted refinement the mesh size scaling tends to 1 2 fairly everywhere on unstructured meshes and geometric consistency is improved with the rectified t spline basis functions the approximation convergence is also greatly enhanced for isogeometric analysis from the original unstructured t splines which is quite meaningful for engineering simulations

a time adaptive finite volume method for the cahn hilliard and kuramoto sivashinsky equations this paper presents a complete finite volume method for the cahn hilliard and kuramoto sivashinsky type of equations the spatial discretization is high order accurate and suitable for general unstructured grids the time integration is addressed by means of implicit an implicit explicit fourth order runge kutta schemes with error control and adaptive time stepping the outcome is a practical accurate and efficient simulation tool which has been successfully applied to accuracy tests and representative simulations the use of adaptive time stepping is of paramount importance in problems governed by the cahn hilliard model an adaptive method may be several orders of magnitude more efficient than schemes using constant or heuristic time steps in addition to driving the simulations efficiently the time adaptive procedure provides a quantitative not just qualitative characterization of the rich temporal scales present in phase separation processes governed by the cahn hilliard phase field model

a discontinuous galerkin method for the cahn hilliard equation a discontinuous galerkin finite element method has been developed to treat the high order spatial derivatives appearing in the cahn hilliard equation the cahn hilliard equation is a fourth order nonlinear parabolic partial differential equation originally proposed to model phase segregation of binary alloys the developed discontinuous galerkin approach avoids the need for mixed finite element methods coupled equations or interpolation functions with a high degree of continuity that have been employed in the literature to treat the fourth order spatial derivatives the variational formulation of the discontinuous galerkin method its implementation and numerical examples are presented in this communication it is also shown under what conditions the method is stable and an error estimate in an energy type norm is presented the method is evaluated by comparison with a standard finite element treatment in which the cahn hilliard equation is decomposed into two coupled partial differential equations