fast methods for the eikonal and related hamilton jacobi equations on unstructured meshes the fast marching method is a numerical algorithm for solving the eikonal equation on a rectangular orthogonal mesh in o m log m steps where m is the total number of grid points the scheme relies on an upwind finite difference approximation to the gradient and a resulting causality relationship that lends itself to a dijkstra like programming approach in this paper we discuss several extensions to this technique including higher order versions on unstructured meshes in rn and on manifolds and connections to more general static hamilton jacobi equations

implicit schemes and lu decompositions implicit methods for hyperbolic equations are analyzed by constructing lu factorizations it is shown that the solution of the resulting tridiagonal systems in one dimension is well conditioned if and only if the lu factors are diagonally dominant stable implicit methods that have diagonally dominant factors are constructed for hyperbolic equations in n space dimesnions only two factors are required even in three space dimensions acceleration to a steady state is analyzed when the multidimensional backward euler method is used with large time steps it is shown that the scheme approximates a newton raphson iteration procedure

numerical analysis of an inverse problem for the eikonal equation we are concerned with the inverse problem for an eikonal equation of determining the speed function using observations of the arrival time on a fixed surface this is formulated as an optimisation problem for a quadratic functional with the state equation being the eikonal equation coupled to the so called soner boundary condition the state equation is discretised by a suitable finite difference scheme for which we obtain existence uniqueness and an error bound we set up an approximate optimisation problem and show that a subsequence of the discrete mimina converges to a solution of the continuous optimisation problem as the mesh size goes to zero the derivative of the discrete functional is calculated with the help of an adjoint equation which can be solved efficiently by using fast marching techniques finally we describe some numerical results

simple finite element based computation of distance functions in unstructured grids a distance field is a representation of the closest distance from a point to a given surface distance fields are widely used in applications ranging from computer vision physics and computer graphics and have been the subject of research of many authors in the last decade most of the methods for computing distance fields are devoted to cartesian grids while little attention has been paid to unstructured grids finite element methods are well known for their ability to deal with partial differential equations in unstructured grids therefore we propose an extension of the fast marching method for computing a distance field in a finite element context employing the element interpolation to hold the eikonal property 1 a simple algorithm to develop the computations is also presented and its efficiency demonstrated through various unstructured grid examples we observed that the presented algorithm has processing times proportional to the number of mesh nodes

topology optimization for turbulent flow with spalart allmaras model abstract in this research a new finite element fe based topology optimization to for turbulent flow was developed using the spalart allmaras s a equation which is one of the reynolds averaged navier stokes rans equations despite many innovative works on the subject of fluidic to it is rare to consider the influence of turbulent flow in to to consider the effect of complex turbulent fluid motion this research considered the s a model developed mainly for aerodynamic flows for a successful to one primary issue is modification of the s a turbulent model to account for topological evolution during the optimization process to address this issue we proposed the addition of penalization terms to the original governing equations to show the validity of the present approach and the effect of turbulent flow on optimal layouts some two dimensional benchmark designs studied for laminar flow were reconsidered by considering the effect of turbulent flow the eddy viscosity values were increased at some local regions due to the boussinesq hypothesis and naturally optimal layouts affected by the spatially varying viscosity were obtained in turbulent flow it was also found that the balance between the inertia force and the viscosity force plays an important role in topological designs

identification for systems with bounded noise in this paper we present a set membership identification algorithm for systems with unknown but bounded disturbance the algorithm contains a weighting factor which is selected according to whether the new observed data contains sufficient information the proposed approach ensures that the estimation error is bounded and nonincreasing furthermore it is shown that the parameter estimates provided by the algorithm will converge to a region containing the true parameters and its upper bound is also given

discrete adjoint based approach for optimization problems on three dimensional unstructured meshes a comprehensive strategy for developing and implementing discrete adjoint methods for aerodynamic shape optimization problems is presented by linearizing each procedure in the entire optimization problem transposing each linearization and reversing the sequential order of operations the adjoint of the complete optimization problem including flow equations and mesh motion equations is constructed in a modular and verifiable fashion this construction is also shown to produce minimal memory overheads and retain the same convergence characteristics of the original analysis problem in the sensitivity analysis these techniques are implemented in a three dimensional unstructured multigrid navierstokes solver and demonstrated on a transonic drag reduction problem for a wing body configuration

robust grid adaptation for efficient uncertainty quantification the meshes to obtain more accurate solutions at each sample point in stochastic space such a procedure can be both cumbersomeandcomputationally expensive toimprove theefficiencyof this process anew robust gridadaptation technique is proposed that is aimed at minimizing the numerical error over a range of variations of the uncertain parameters of interest about a nominal state using this approach it is possible to generate computational grids that are insensitive to small variations of the uncertain parameters that can both locally and globally change the solution and as a result the error distribution this is in contrast with classical adjoint techniques which seek to adapt the gridwiththeaimofminimizingnumericalerrorsforaspecific flowcondition andgeometry itisdemonstratedthat flow computations on these robust grids result in low numerical errors under the expected range of variations of the uncertain input parameters the effectiveness of this strategy is demonstrated in problems involving the poisson equation and the euler equations at transonic and supersonic hypersonic speeds

continuous adjoint approach for the spalart allmaras model in aerodynamic optimization a new variable solving the eikonal equation the accuracy of the sensitivity derivatives obtained with the complete turbulent approach is assessed by comparison with finite difference computations and the classical continuous adjoint with frozen viscosity showing substantial improvements in the convergence properties of the method and in the quality of the obtained gradients the validity of the overall methodology is illustrated with several design examples including the optimization of three dimensional geometries in combination with advanced freeform techniques for mesh deformation

adjoint consistency analysis of discontinuous galerkin discretizations this paper is concerned with the adjoint consistency of discontinuous galerkin dg discretizations adjoint consistency in addition to consistency is the key requirement for dg discretizations to be of optimal order in l 2 as well as measured in terms of target functionals we provide a general framework for analyzing the adjoint consistency of dg discretizations which is also useful for the derivation of adjoint consistent methods this analysis will be performed for the dg discretizations of the linear advection equation the interior penalty dg method for elliptic problems and the dg discretization of the compressible euler equations this framework is then used to derive an adjoint consistent dg discretization of the compressible navier stokes equations numerical experiments demonstrate the link of adjoint consistency to the accuracy of numerical flow solutions and the smoothness of discrete adjoint solutions

effect of approximations of the discrete adjoint on gradient based optimization an exact discrete adjoint of an unstructured nite volume solver for the rans equations has been developed the adjoint is exact in the sense of being based on the full linearization of all terms in the solver including all turbulence model contributions from this starting point various approximations to the adjoint are derived with the intention of simplifying the development and memory requirements of the method considered are many approximations already seen in the literature the eect of these approximations on the accuracy of the resulting design gradients and the convergence and nal solution of optimizations is studied as it applies to a two dimensional high lift conguration

superconvergent functional estimates from summation by parts finite difference discretizations diagonal norm summation by parts sbp operators can be used to construct time stable high order accurate finite difference schemes however to achieve both stability and accuracy these operators must use s order accurate boundary closures when the interior scheme is 2s order accurate the boundary closure limits the solution to s 1 order global accuracy despite this bound on solution accuracy we show that functional estimates can be constructed that are 2s order accurate this superconvergence requires dual consistency which depends on the sbp operators the boundary condition implementation and the discretized functional the theory is developed for scalar hyperbolic and elliptic partial differential equations in one dimension in higher dimensions we show that superconvergent functional estimates remain viable in the presence of curvilinear multiblock grids with interfaces the generality of the theoretical results is demonstrated using a two dimensional poisson problem and a nonlinear hyperbolic system the euler equations of fluid mechanics

convergence of linearized and adjoint approximations for discontinuous solutions of conservation laws part 1 linearized approximations and linearized output functionals this paper analyzes the convergence of discrete approximations to the linearized equations arising from an unsteady one dimensional hyperbolic equation with a convex flux function a simple modified lax friedrichs discretization is used on a uniform grid and a key point is that the numerical smoothing increases the number of points across the nonlinear discontinuity as the grid is refined it is proved that this gives pointwise convergence almost everywhere for the solution to the linearized discrete equations with smooth initial data and also convergence in the discrete approximation of linearized output functionals in part 2 m giles and s ulbrich siam j numer anal 48 2010 pp 905 921 we extend the results to dirac initial data for the linearized equation and will prove the pointwise convergence almost everywhere for the solution of the adjoint discrete equations in addition we present numerical results illustrating the asymptotic behavior which is analyzed

optimal control of the propagation of a graph in inhomogeneous media we study an optimal control problem for viscosity solutions of a hamilton jacobi equation describing the propagation of a one dimensional graph with the control being the speed function the existence of an optimal control is proved together with an approximate controllability result in the h 1 norm we prove convergence of a discrete optimal control problem based on a monotone finite difference scheme and describe some numerical results

adjoint an approach for the rapid development of discrete adjoint solvers an automatic differentiation tool is used to develop the adjoint code for a three dimensional computational fluid dynamics solver rather than using automatic differentiation to differentiate the entire source code of the computational fluid dynamics solver we have applied it selectively to produce code that computes the flux jacobian matrix and the other partial derivatives that are necessary to compute total derivatives using an adjoint method the resulting linear discrete adjoint system is then solved using the portable extensible toolkit for scientific computation this selective application of automatic differentiation is the central idea behind the automatic differentiation adjoint adjoint approach this approach has the advantage that it is applicable to arbitrary sets of governing equations and cost functions and that it is exactly consistent with the gradients that would be computed by exact numerical differentiation of the original solver furthermore the approach is largely automatic thus avoiding the lengthy development times usually required to develop adjoint solvers for partial differential equations these significant advantages come at the cost of increased memory requirements for the adjoint solver derivatives of drag and lift coefficients are validated and the low computational cost and ease of implementation of the method are shown

convergence of linearized and adjoint approximations for discontinuous solutions of conservation laws part 2 adjoint approximations and extensions this paper continues the convergence analysis in m giles and s ulbrich siam j numer anal 48 2010 pp 882 904 of discrete approximations to the linearized and adjoint equations arising from an unsteady one dimensional hyperbolic equation with a convex flux function we consider a simple modified lax friedrichs discretization on a uniform grid and a key point is that the numerical smoothing increases the number of points across the nonlinear discontinuity as the grid is refined it is proved that there is convergence in the discrete approximation of linearized output functionals even for dirac initial perturbations and pointwise convergence almost everywhere for the solution of the adjoint discrete equations in particular the adjoint approximation converges to the correct uniform value in the region in which characteristics propagate into the discontinuity moreover it is shown that the results of m giles and s ulbrich siam j numer anal 48 2010 pp 882 904 and the present paper hold also for quite general nonlinear initial data which contain multiple shocks and for which shocks form at a later time and or merge

systematic continuous adjoint approach to viscous aerodynamic design on unstructured grids doi 10 2514 1 24859 a continuous adjoint approach to aerodynamic design for viscous compressible flow on unstructured grids is developed sensitivity gradients arecomputed using tools ofshapedeformation of boundaryintegrals theresulting expressions involve second order derivatives of the flow variables that require numerical solvers with greater than second order accuracy a systematic way of reducing the order of these terms is presented the accuracy of the sensitivity derivatives is assessed by comparison with finite difference computations and the validity of the overall methodology is illustrated with several design examples

the ball vertex method a new simple spring analogy method for unstructured dynamic meshes we study the problem of deforming unstructured grids using pseudo structural lumped parameter systems a basic network of edge springs is complemented with an additional set of linear springs that oppose element collapsing this is here achieved by confining each mesh vertex to its ball through linear springs that are attached to the vertex and to its projections on the ball entities the resulting linear pseudo structural problem can be solved efficiently with an iterative method the proposed procedures are compared with a revisited version of the torsional spring method with the help of two and three dimensional example problems

an a priori anisotropic goal oriented error estimate for viscous compressible flow and application to mesh adaptation abstract we present a goal oriented error analysis for the calculation of low reynolds steady compressible flows with anisotropic mesh adaptation the error analysis is of a priori type its central principle is to express the right hand side of the error equation often referred as the local error as a function of the interpolation error of a collection of fields present in the nonlinear partial differential equations this goal oriented error analysis is the extension of 39 done for inviscid flows to laminar viscous flows by adding viscous terms the main benefits of this approach in comparison to other error estimates in the literature is that the optimal anisotropy of the mesh directly appears in the error analysis and is not obtained from an ad hoc variable nor a local analysis as a consequence an optimum is obtained and the convergence of the mesh adaptive process is very fast i e generally the convergence is obtained after 5 to 10 mesh adaptation cycle then using the continuous mesh framework an optimal metric is analytically obtained from the error estimation applications to mesh adaptive calculations of flows past airfoils are presented

graphical perception theory experimentation and application to the development of graphical methods abstract the subject of graphical methods for data analysis and for data presentation needs a scientific foundation in this article we take a few steps in the direction of establishing such a foundation our approach is based on graphical perception the visual decoding of information encoded on graphs and it includes both theory and experimentation to test the theory the theory deals with a small but important piece of the whole process of graphical perception the first part is an identification of a set of elementary perceptual tasks that are carried out when people extract quantitative information from graphs the second part is an ordering of the tasks on the basis of how accurately people perform them elements of the theory are tested by experimentation in which subjects record their judgments of the quantitative information on graphs the experiments validate these elements but also suggest that the set of elementary tasks should be expanded the theory provides a guideline for graph construction

on graphical tests for proportionality of hazards in two samples in this paper we present a class of graphical tests of the proportional hazards hypothesis for two sample censored survival data the proposed tests are improvements over some existing tests based on asymptotic confidence bands of certain functions of the estimated cumulative hazard functions the new methods are based on the comparison of unrestricted estimates of the said functions and their restricted versions under the hypothesis they combine the rigour of analytical tests with the descriptive value of plots monte carlo simulations suggest that the proposed asymptotic procedures have reasonable small sample properties the power is much higher than existing graphical tests and comparable with existing analytical tests the method is then illustrated through the analysis of a data set on bone marrow transplantation for leukemia patients