grid generation techniques in computational fluid dynamics revue des differentes methodes on montre que les regions tridimensionnelles sont mieux traitees par fragmentation du champ en sous regions avec generation de grilles dans chaque sous region interet des grilles adaptatives couplees a des solutions physiques

a mixed variable continuously deforming finite element method for parabolic evolution problems part i the variational formulation for a single evolution equation the objective of the research presented here was to develop a generic adaptive computational method for porous media evolution problems that involve coupled heat flow fluid flow and species transport processes with sharply defined phase change interfaces in this paper we examine the general least squares variational approach and develop the conceptual framework for a rate least squares variational formulation of a continuously deforming mixed variable finite element method for solving highly non linear time dependent partial differential equations in part ii of this paper1 we extend the formulation given here for a single evolution equation to a system of coupled evolution equations in part iii2 we discuss in detail the numerical procedures that were implemented in a computer program and present several numerical examples that demonstrate the performance of this computational method

bayesian methods for pharmacokinetic models in dynamic contrast enhanced magnetic resonance imaging this paper proposes a new method for estimating kinetic parameters of dynamic contrast enhanced magnetic resonance imaging dce mri based on adaptive gaussian markov random fields kinetic parameter estimates using neighboring voxels reduce the observed variability in local tumor regions while preserving sharp transitions between heterogeneous tissue boundaries asymptotic results for standard errors from likelihood based nonlinear regression are compared with those derived from the posterior distribution using bayesian estimation with and without neighborhood information application of the method to the analysis of breast tumors based on kinetic parameters has shown that the use of bayesian analysis combined with adaptive gaussian markov random fields provides improved convergence behavior and more consistent morphological and functional statistics

continuously deforming finite elements a general class of time dependent co ordinate transformations is introduced in a variational formulation for evolution problems the variational problem is posed with respect to both solution and transformation field variables an approximate analysis using finite elements is developed from the continuous variational form modified forms of the variational functional are considered to ensure the deforming mesh is not top irregular ode system integrators are utilized to integrate the resulting semidiscrete systems in part i we consider the formulation for problems in one spatial dimension and time including in particular convection dominated flows described by the convection diffusion burgers and buckley leverett equations in part ii the extension of the method to two dimensions and supporting numerical experiments are presented

mesh independent spectra in the moving finite element equations we derive the moving finite element mfe equations for the solution of a scalar evolutionary equation in d space dimensions d geqq 1 and introduce the elementwise approach to mfe this approa

boundary fitted coordinate systems for numerical solution of partial differential equations a review a comprehensive review of methods of numerically generating curvilinear coordinate systems with coordinate lines coincident with all boundary segments is given some general mathematical framework and error analysis common to such coordinate systems is also included the general categories of generating systems are those based on conformal mapping orthogonal systems nearly orthogonal systems systems produced as the solution of elliptic and hyperbolic partial differential equations and systems generated algebraically by interpolation among the boundaries also covered are the control of coordinate line spacing by functions embedded in the partial differential operators of the generating system and by subsequent stretching transformation dynamically adaptive coordinate systems coupled with the physical solution and time dependent systems that follow moving boundaries are treated references reporting experience using such coordinate systems are reviewed as well as those covering the system development

k version of finite element method in gas dynamics higher order global differentiability numerical solutions in this paper we consider and examine alternate finite element computational strategies for time dependent navier stokes equations describing high speed compressible flows with shocks in a viscous and conducting medium with the ultimate objective of establishing the desired features of a general mathematical and computational framework for such initial value problems ivp in which a the numerically computed solutions are in agreement with the physics of evolution described by the governing differential equations gdes i e the ivp b the solutions are admissible in the non discretized form of the gdes in the pointwise sense i e anywhere and everywhere in the entire space time domain and hence in the integrated sense as well c the numerical approximations progressively approach the same global differentiability in space and time as the theoretical solutions d it is possible to time march the solutions this is essential for efficiency as well as ensuring desired accuracy of the computed solution for the current increment of time i e to minimize the error build up in the time marching process e the computational process is unconditionally stable and non degenerate regardless of the choice of discretization nature of approximations and their global differentiability and the dimensionless parameters influencing the physics of the process f there are no issues of stability cfl number limitations and g the mathematical and computational methodology is independent of the nature of the space time differential operators we consider one dimensional compressible flow in a viscous and conducting medium with shocks as model problems to illustrate various features of the general mathematical and computational framework used here and to demonstrate that the proposed framework is general and is applicable to all ivp the riemann shock tube with a single diaphragm serves as a model problem the specific details presented in the paper discuss 1 choice of the form of the gdes i e strong form or weak form 2 various choices of variables the paper establishes and considers density velocity and temperature as variables of choice 3 details of the space time least squares ls integral forms meritorious over all others in all aspects are presented and choice of approximation spaces are discussed 4 in all numerical studies we consider a viscous and conducting medium with ideal gas law however results are also presented for non conducting medium extension of this work to real gas models will be presented in a separate paper it is worth noting that when the medium is viscous and conducting the solutions of gas dynamics equations are analytic 5 it is also significant to note that upwinding methods based on addition of artificial diffusion such as supg supg dc supg dc ls and their many variations are neither needed nor used in this present work 6 numerical studies are aimed at resolving the localized details of the shock structure i e shock relations shock width shock speed etc as well as the over all global behaviour of the solution in the entire space time domain 7 numerical studies are presented for riemann shock tube for high mach number flows with special emphasis also on time accuracy of the evolution which is ensured by requiring that the approximations for each increment of time satisfy non discretized form of the gdes in the pointwise sense and hence in the integrated sense as well 8 comparisons are made with published results as well as theoretical solutions when possible it is established that space time least squares processes are the only processes that yield variationally consistent space time integral forms and hence unconditionally non degenerate space time computational processes which when considered in higher order scalar product spaces provide the desired mathematical framework in which progressively higher order global differentiability solutions in space and time yield the same characteristics as the theoretical solutions of the ivp in all aspects copyright 2006 john wiley sons ltd

a moving petrov galerkin method for transport equations a time dependent transport equation is solved numerically by using a petrov galerkin technique on a time dependent grid a piecewise linear trial solution and piecewise hermite cubic test functions are used the same method is also applied to a time independent problem in which case it automatically generates a grid in a way which bears some resemblance to certain existing grid generation techniques several numerical examples are given

p version least squares finite element formulation for convection diffusion problems this paper presents a p version least squares finite element formulation of the convection diffusion equation the second order differential equation describing convection diffusion is reduced to a series of equivalent first order differential equations for which the least squares formulation is constructed using the same order of approximation for each of the dependent variables the hierarchical approximation functions and the nodal variable operators are established by first constructing the one dimensional hierarchical approximation functions of orders p and p and the corresponding nodal variable operators in and direction and then taking their products numerical results are presented and compared with analytical and numerical solutions for a two dimensional test problem to demonstrate the accuracy and the convergence characteristics of the present formulation the gaussian quadrature rule used to calculate the numerical values of the element matrices vectors as well as the error functional i e is established based on the highest degree of the polynomial in the integrands it is demonstrated that this quadrature rule with the present p version formulation produces excellent results for very low as well as extremely high peclet numbers 10 106 and furthermore the error functional i sum of the integrals of e2 is a monotonically decreasing function of the number of degrees of freedom as the p levels are increased for a fixed mesh it is shown that exact integration with the h version linear and parabolic elements produces inaccurate solutions at high peclet numbers results are also presented using reduced integration it is shown that the reduced integration with p version produces accurate values of the primary variable even for relatively low p levels but the error functional i when calculated using the proper integration rule has a much higher value due to errors in the derivatives of the primary variable and is a non monotonic function of the degrees of freedom as p levels are increased for a fixed mesh similar behaviour of the error functional i is also observed for the h models using linear elements when reduced integration is used although the h models using parabolic elements produce monotonic error functional behaviour as the number of degrees of freedom are increased the error values are inferior to the p version results using exact integration

equidistributing principles in moving finite element methods abstract recently miller and his co workers proposed a moving finite element method based on a least squares principle this was followed by a similar method by the present authors using a petrov galerkin approach in this paper the two methods are compared in particular it is shown that both methods move their nodes according to an approximate equidistributing principle this observation leads to a criterion for the placement of the nodes it is also shown that the penalty function designed by miller may also be used with the petrov galerkin method finally numerical examples are given illustrating the performance of the two methods

p version least squares finite element formulation for axisymmetric incompressible non newtonian fluid flow abstract the paper presents a p version least squares finite element formulation lsfef for axisymmetric incompressible non newtonian fluid flow the dimensionless form of the equations describing the fluid motion are cast into a set of first order coupled partial differential equations involving non newtonian stresses as auxiliary variables the pressure velocities primary variables and the stresses auxiliary variables are interpolated using equal order c0 continuity p version hierarchical approximation functions the least squares functional or error functional is constructed using the system of coupled first order non linear partial differential equations integrated sum of squares of the errors resulting from the individual equations for the entire discretization without linearization approximations or assumptions the minimization of this least squares error functional results in finding a solution vector for which the partial derivatives of the error functional with respect to the nodal degrees of freedom becomes zero this is accomplished by using newton s method with a line search the paper presents an implementation of the power law model for non newtonian viscosity numerical examples are presented to demonstrate the convergence characteristics and the accuracy of the proposed formulation

moving finite elements i we present new and general numerical methods for dealing with problems whose solutions develop sharp transition layers or near shocks these methods allow many nodes automatically to concentrate in the critical regions and move with them for clarity of exposition we concentrate on the space of piecewise linear functions with movable nodes with burgers equation as our test equation but the generalization to much more general spaces and equations including even certain previous moving vorticity blobs of the first author and s doss for the navier stokes equations becomes clear in this paper we present the theoretical and computational details of our scheme along with computational trial runs for burgers equation showing that the nodes do concentrate sharply and move with the shocks as desired the conclusiveness of these preliminary numerical trials is marred somewhat by the fact that we never successfully debugged a newton s method for our implicit stiff ode solver and were thus limited to ver

a mixed variable continuously deforming finite element method for parabolic evolution problems part ii the coupled problem of phase change in porous media the conceptual framework of a least squares rate variational approach to the formulation of continuously deforming mixed variable finite element computational scheme for a single evolution equation was presented in part i 1 in this paper part ii we extend these concepts and present an adaptively deforming mixed variable finite element method for solving general two dimensional transport problems governed by a system of coupled non linear partial differential evolution equations in particular we consider porous media problems that involve coupled heat and mass transport processes that yield steep continuous moving fronts and abrupt discontinuous moving phase change interfaces in this method the potentials such as the temperature pressure and species concentration and the corresponding fluxes are permitted to jump in value across the phase change interfaces the equations and the jump conditions governing the physical phenomena which were specialized from a general multiphase multiconstituent mixture theory provided the basis for the development and implementation of a two dimensional numerical simulator this simulator can effectively resolve steep continuous fronts i e shock capturing without oscillations or numerical dispersion and can accurately represent and track discontinuous fronts i e shock fitting through adaptive grid deformation and redistribution the numerical implementation of this simulator and numerical examples that demonstrate the performance of the computational method are presented in part iii2 of this paper

a space time coupled p version least squares finite element formulation for unsteady two dimensional navier stokes equations this paper presents a p version least squares finite element formulation for two dimensional unsteady fluid flow described by navier stokes equations where the effects of space and time are coupled the dimensionless form of the navier stokes equations are first cast into a set of first order differential equations by introducing auxiliary variables this permits the use of c0 element approximation the element properties are derived by utilizing the p version approximation functions in both space and time and then minimizing the error functional given by the space time integral of the sum of squares of the errors resulting from the set of first order differential equations this results in a true space time coupled least squares minimization procedure the application of least squares minimization to the set of coupled first order partial differential equations results in finding a solution vector which makes gradient of error functional with respect to a null vector this is accomplished by using newton s method with a line search a time marching procedure is developed in which the solution for the current time step provides the initial conditions for the next time step equilibrium iterations are carried out for each time step until the error functional and each component of the gradient of the error functional with respect to nodal degrees of freedom are below a certain prespecified tolerance the space time coupled p version approximation functions provide the ability to control truncation error which in turn permits very large time steps what literally requires hundreds of time steps in uncoupled conventional time marching procedures can be accomplished in a single time step using the present space time coupled approach the generality success and superiority of the present formulation procedure is demonstrated by presenting specific numerical examples for transient couette flow and transient lid driven cavity the results are compared with the analytical solutions and those reported in the literature the formulation presented here is ideally suited for space time adaptive procedures the element error functional values provide a mechanism for adaptive h p or hp refinements

generalized orthopair fuzzy sets we note that orthopair fuzzy subsets are such that that their membership grades are pairs of values from the unit interval one indicating the degree of support for membership in the fuzzy set and the other support against membership we discuss two examples atanassov s classic intuitionistic sets and a second kind of intuitionistic set called pythagorean we note that for classic intuitionistic sets the sum of the support for and against is bounded by one while for the second kind pythagorean the sum of the squares of the support for and against is bounded by one here we introduce a general class of these sets called q rung orthopair fuzzy sets in which the sum of the rm q th power of the support for and the rm q th power of the support against is bonded by one we note that as q increases the space of acceptable orthopairs increases and thus gives the user more freedom in expressing their belief about membership grade we investigate various set operations as well as aggregation operations involving these types of sets

a space time coupled p version least squares finite element formulation for unsteady fluid dynamics problems this paper presents a p version least squares finite element formulation for unsteady fluid dynamics problems where the effects of space and time are coupled the dimensionless form of the differential equations describing the problem are first cast into a set of first order differential equations by introducing auxiliary variables this permits the use of c element approximation the element properties are derived by utilizing p version approximation functions in both space and time and then minimizing the error functional given by the space time integral of the sum of squares of the errors resulting from the set of first order differential equations this results in a true space time coupled least squares minimization procedure a time marching procedure is developed in which the solution for the current time step provides the initial conditions for the next time step the space time coupled p version approximation functions provide the ability to control truncation error which in turn permits very large time steps what literally requires hundreds of time steps in uncoupled conventional time marching procedures can be accomplished in a single time step using the present space time coupled approach for non linear problems the non linear algebraic equations resulting from the least squares process are solved using newton s method with a line search this procedure results in a symmetric hessian matrix equilibrium iterations are carried out for each time step until the error functional and each component of the gradient of the error functional with respect to nodal degrees of freedom are below a certain prespecified tolerance the generality success and superiority of the present formulation procedure is demonstrated by presenting specific formulations and examples for the advection diffusion and burgers equations the results are compared with the analytical solutions and those reported in the literature the formulation presented here is ideally suited for space time adaptive procedures the element error functional values provide a mechanism for adaptive h p or hp refinements the work presented in this paper provides the basis for the extension of the space time coupled least squares minimization concept to two and three dimensional unsteady fluid flow

on the statistical analysis of dirty pictures may 7th 1986 professor a f m smith in the chair summary a continuous two dimensional region is partitioned into a fine rectangular array of sites or pixels each pixel having a particular colour belonging to a prescribed finite set the true colouring of the region is unknown but associated with each pixel there is a possibly multivariate record which conveys imperfect information about its colour according to a known statistical model the aim is to reconstruct the true scene with the additional knowledge that pixels close together tend to have the same or similar colours in this paper it is assumed that the local characteristics of the true scene can be represented by a nondegenerate markov random field such information can be combined with the records by bayes theorem and the true scene can be estimated according to standard criteria however the computational burden is enormous and the reconstruction may reflect undesirable largescale properties of the random field thus a simple iterative method of reconstruction is proposed which does not depend on these large scale characteristics the method is illustrated by computer simulations in which the original scene is not directly related to the assumed random field some complications including parameter estimation are discussed potential applications are mentioned briefly

complex fuzzy logic a novel framework for logical reasoning termed complex fuzzy logic is presented in this paper complex fuzzy logic is a generalization of traditional fuzzy logic based on complex fuzzy sets in complex fuzzy logic inference rules are constructed and fired in a manner that closely parallels traditional fuzzy logic the novelty of complex fuzzy logic is that the sets used in the reasoning process are complex fuzzy sets characterized by complex valued membership functions the range of these membership functions is extended from the traditional fuzzy range of 0 1 to the unit circle in the complex plane thus providing a method for describing membership in a set in terms of a complex number several mathematical properties of complex fuzzy sets which serve as a basis for the derivation of complex fuzzy logic are reviewed in this paper these properties include basic set theoretic operations on complex fuzzy sets namely complex fuzzy union and intersection complex fuzzy relations and their composition and a novel form of set aggregation vector aggregation complex fuzzy logic is designed to maintain the advantages of traditional fuzzy logic while benefiting from the properties of complex numbers and complex fuzzy sets the introduction of complex valued grades of membership to the realm of fuzzy logic generates a framework with unique mathematical properties and considerable potential for further research and application

spatially regularized estimation for the analysis of dynamic contrast enhanced magnetic resonance imaging data competing compartment models of different complexities have been used for the quantitative analysis of dynamic contrast enhanced magnetic resonance imaging data we present a spatial elastic net approach that allows to estimate the number of compartments for each voxel such that the model complexity is not fixed a priori a multi compartment approach is considered which is translated into a restricted least square model selection problem this is done by using a set of basis functions for a given set of candidate rate constants the form of the basis functions is derived from a kinetic model and thus describes the contribution of a specific compartment using a spatial elastic net estimator we chose a sparse set of basis functions per voxel and hence rate constants of compartments the spatial penalty takes into account the voxel structure of an image and performs better than a penalty treating voxels independently the proposed estimation method is evaluated for simulated images and applied to an in vivo dataset copyright 2013 john wiley sons ltd

three validation metrics for automated probabilistic image segmentation of brain tumours the validity of brain tumour segmentation is an important issue in image processing because it has a direct impact on surgical planning we examined the segmentation accuracy based on three two sample validation metrics against the estimated composite latent gold standard which was derived from several experts manual segmentations by an em algorithm the distribution functions of the tumour and control pixel data were parametrically assumed to be a mixture of two beta distributions with different shape parameters we estimated the corresponding receiver operating characteristic curve dice similarity coefficient and mutual information over all possible decision thresholds based on each validation metric an optimal threshold was then computed via maximization we illustrated these methods on mr imaging data from nine brain tumour cases of three different tumour types each consisting of a large number of pixels the automated segmentation yielded satisfactory accuracy with varied optimal thresholds the performances of these validation metrics were also investigated via monte carlo simulation extensions of incorporating spatial correlation structures using a markov random field model were considered

a mixed variable continuously deforming finite element method for parabolic evolution problems part iii numerical implementation and computational results in part ii a system of ordinary differential equations with respect to time was derived for solving a system of three coupled evolution equations by the deforming grid mixed variable least squares rate variational finite element method in this paper we present the method we have adopted to integrate with respect to time the resulting spatially discretized system of non linear differential equations we present computational results obtained using the code in which this deforming mixed finite element method was implemented the problems were selected to exercise these features with the objective of demonstrating that the formulation is correct and that the numerical procedures adopted converge to the correct solutions

estimating kinetic parameter maps from dynamic contrast enhanced mri using spatial prior knowledge dynamic contrast enhanced magnetic resonance dce mr imaging can be used to study microvascular structure in vivo by monitoring the abundance of an injected diffusible contrast agent over time the resulting spatially resolved intensity time curves are usually interpreted in terms of kinetic parameters obtained by fitting a pharmacokinetic model to the observed data least squares estimates of the highly nonlinear model parameters however can exhibit high variance and can be severely biased as a remedy we bring to bear spatial prior knowledge by means of a generalized gaussian markov random field ggmrf by using information from neighboring voxels and computing the maximum a posteriori solution for entire parameter maps at once both bias and variance of the parameter estimates can be reduced thus leading to smaller root mean square error rmse since the number of variables gets very big for common image resolutions sparse solvers have to be employed to this end we propose a generalized iterated conditional modes icm algorithm operating on blocks instead of sites which is shown to converge considerably faster than the conventional icm algorithm results on simulated dce mr images show a clear reduction of rmse and variance as well as in some cases reduced estimation bias the mean residual bias mrb is reduced on the simulated data as well as for all 37 patients of a prostate dce mri dataset using the proposed algorithm average computation times only increase by a factor of 1 18 871 ms per voxel for a gaussian prior and 1 51 1 12 s per voxel for an edge preserving prior compared to the single voxel approach 740 ms per voxel

comment on pythagorean and complex fuzzy set operations dick et al introduced a relation leq on pi i numbers and proved that pi i leq is a bounded distributive complete lattice in the present paper we point out that the relation leq is not a partial order and that pi i leq is not a lattice in addition we study some partial orders on pi i numbers