efficient merging of multiple segments of bezier curves this paper deals with the merging problem of segments of a composite bezier curve with the endpoints continuity constraints we present a novel method which is based on the idea of using constrained dual bernstein polynomial basis wo ny and lewanowicz 2009 12 to compute the control points of the merged curve thanks to using fast schemes of evaluation of certain connections involving bernstein and dual bernstein polynomials the complexity of our algorithm is significantly less than complexity of other merging methods

differential recurrence properties of dual bernstein polynomials abstract new differential recurrence properties of dual bernstein polynomials are given which follow from relations between dual bernstein and orthogonal hahn and jacobi polynomials using these results a fourth order differential equation satisfied by dual bernstein polynomials has been constructed also a fourth order recurrence relation for these polynomials has been obtained this result may be useful in the efficient solution of some computational problems

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

constrained degree reduction of polynomials in bernstein bezier form over simplex domain in this paper we show that the orthogonal complement of a subspace in the polynomial space of degree n over d dimensional simplex domain with respect to the l 2 inner product and the weighted euclidean inner product of bb bezier bernstein coefficients are equal using it we also prove that the best constrained degree reduction of polynomials over the simplex domain in bb form equals the best approximation of weighted euclidean norm of coefficients of given polynomial in bb form from the coefficients of polynomials of lower degree in bb form

three dimensional grid generation using elliptic equations with direct grid distribution control boundary fitted computational meshes are generated by solving a system of elliptic partial differential equations the grid points at the boundary surfaces are arbitrarily distributed the interior grid distribution is controlled by a technique which insures a grid line orthogonality at the boundaries the grid is automatically adjusted while the grid line orthogonality at the boundaries is achieved in an iterative process this technique is demonstrated for three different configurations desirable cell size and shape are obtained for the boundary shapes which normally cause poor grid distribution

a non iterative scheme for orthogonal grid generation with control function and specified boundary correspondence on three sides a new numerical scheme is proposed for generating an orthogonal grid in a simply connected 2d domain the scheme is based on the idea of decomposition of a global orthogonal transform into consecutive mappings of a conformal mapping and an auxiliary orthogonal mapping which was suggested by kang and leal j comput phys 102 78 1992 the method is non iterative and flexible in the adjustment of grid spacing the grid spacing can be controlled mainly by specification of the boundary correspondence up to three sides of the boundary the method is also equipped with a control function that provides further degrees of freedom in the grid spacing adjustment from a mathematical viewpoint the proposed scheme can also be regarded as a numerical implementation of the constructive proof for the existence of a solution of the orthogonal mapping problem in an arbitrary simply connected domain under the condition that the boundary correspondence is specified on three sides

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

advantageous strategies in differential games for a class of suboptimal control laws abstract this paper deals with suboptimal control policies in differential games sufficient conditions for advantageous strategies of either player are examined for a linear quadratic game in which the players are constrained to use suboptimal control laws of specified form the concept of a bargaining matrix is employed specific scalar and vector examples are included to illustrate the theory

simple algorithms for computing the bezier coefficients of the constrained dual bernstein polynomials abstract simple recurrence relations for the bezier coefficients of the dual bernstein polynomials are given using the recurrence relations we propose fast methods for computing these coefficients the complexity of the proposed algorithms is o n where n is the degree of the considered dual bernstein polynomial basis

merging of bezier curves with box constraints in this paper we present a novel approach to the problem of merging of bezier curves with respect to the l 2 norm we give illustrative examples to show that the solution of the conventional merging problem may not be suitable for further modification and applications as in the case of the degree reduction problem we apply the so called restricted area approach proposed recently in gospodarczyk 2015 to avoid certain defects and make the resulting curve more useful a method of solving the new problem is based on box constrained quadratic programming approach we present a novel approach to the problem of merging of bezier curves we impose constraints of a new type we give illustrative examples to explain the purpose of the new restrictions the merged curve is suitable for further modification and applications

control point based exact description of trigonometric hyperbolic curves surfaces and volumes using the normalized b bases of vector spaces of trigonometric and hyperbolic polynomials of finite order we specify control point configurations for the exact description of the zeroth and higher order mixed partial derivatives of integral curves and hybrid multivariate surfaces determined by coordinate functions that are exclusively given either by traditional trigonometric or hyperbolic polynomials in each of their variables based on homogeneous coordinates and central projection we also propose algorithms for the control point and weight based exact description of the zeroth order partial derivative of the rational counterpart of these integral curves and surfaces the core of the proposed modeling methods relies on basis transformation matrices with entries that can be efficiently obtained by order elevation

approximation of parametric curves by moving least squares method in this work we propose a method to approximate a parametric curve in r d some distinct points in r d are given we assume that these points belong to a parametric curve and our aim is to approximate these data by moving least squares method we mention several applications of the proposed method to emphasize the importance of the work also root mean squares errors and hausdorff distances between the exact curve and its approximation are presented to demonstrate the efficiency and reliability of the method

an initial value approach to the production of discrete orthogonal coordinates abstract a numerical procedure is described for the orthogonalization of a discrete non orthogonal coordinate system in two dimensions one set of the non orthogonal coordinate lines is preserved by the orthogonalization procedure the other set is shifted by interpolation to form the new orthogonal system the procedure is based upon an initial value formulation with the non orthogonal coordinates as the independent variables a first order linear partial differential equation is derived which describes the orthogonal trajectories a discrete representation of the equation is suggested and an interpolation procedure is described examples of non orthogonal with orthogonalized meshes are given including closed and open coordinate lines the method may be used to generate boundary fitted coordinates for boundary value problems or eulerian time dependent codes it may also be used at each timestep in a waterbag or semi lagrangian code

orthogonal grid generation in a 2d domain via the boundary integral technique a new numerical scheme is proposed for the generation of an orthogonal coordinate grid in an arbitrary simply connected two dimensional domain the scheme is robust and non iterative and is based on the conjunction of the familiar boundary integral technique with the covariant laplace equation method for mapping in the proposed scheme two types of problems are considered 1 boundary correspondence is specified on two adjacent sides of the boundary or 2 the distortion factor is specified in the product form f x h x q h

an explicit method for g 3 merging of two bezier curves this paper presents an explicit method for the g 3 merging problem of two bezier curves the main idea is to express the l 2 distance as a quadratic function of some parameters provided by g 3 continuity conditions an efficient non iterative algorithm is proposed to obtain the optimal merged curve when the l 2 distance is minimized the uniqueness of the global minimum is also proven this method can be applied to two adjacent curves with different degrees and has the ability to obtain satisfactory merging results by using curves of lower degree the efficiency and accuracy of the proposed explicit method are illustrated through several comparative examples

explicit g2 constrained degree reduction of bezier curves by quadratic optimization in this paper we revisit g 2 constrained degree reduction of bezier curves which has been solved in our previous work by using iterative methods we propose an explicit and effective method for g 1 constrained degree reduction and c 1g 2 constrained degree reduction our main idea is to express the distance function defined in the l 2 norm as a strictly convex quadratic function of two variables which becomes a quadratic optimization problem we can explicitly obtain the unique solution by solving two linear equations such that the distance function is minimized the existence of the unique solution is also proved

orthogonal grids on meander like regions we deal with the automatic construction of orthogonal grids on meander like regions these are similar to riverbeds and appear naturally in geography rivers regions between two fixed heights in a hilly landscape and medicine longitudinal slices of organs such as veins and arteries we use pairs of lemniscates of two foci and pairs of ellipses to approximate subregions of the meander like region a subregion determines an approximating lemniscatic or elliptic sector given by two confocal lemniscates or two ellipses from a certain family and in both cases two orthogonal hyperbolas we use the fact that a sector is conformally equivalent to a region bounded by two concentric circles and two rays to map the orthogonal grid on this disc domain given by polar coordinates into a grid in the approximating sector finally the lemniscatic and elliptic sectors that approximate the subregions are assembled to conform an orthogonal grid on the meander like region

a laplacian equation method for numerical generation of boundary fitted 3d orthogonal grids abstract a sethod for generating boundary fitted orthogonal curvilinear grids in 3 dimensional space is described the mapping between the curvilinear coordinates and the cartesian coordinates is provided by a set of laplace equations which expressed in curvilinear coordinates involve the components of the metric tensor and are therefore non linear and coupled an iterative algorithm is described which achieves a numerical solution grids appropriate for the calculation of flow fields over complex topography or in complex flow passages as those found in turbomachinery and for other engineering applications can be constructed using the proposed method various examples are presented and plotted in perspective and data for the assessment of the properties of the resulting meshes is provided

the numerical solution of three stochastic differential games abstract in this chapter stochastic versions of three differential games namily the blocking game 1 chap 6 the game of patrolling channel 1 chap 9 and the collision avoidance game 2 chap 8 are considered the common factor of these differential games is that all three of them are modelled by the same set of stochastic differential equations in each of the games sufficient conditions on optimal strategies in the case of complete observation and sufficient conditions on weak optimal strategies in the case of partial observations are derived in the cases of complete observation optimal strategies are computed for a variety of cases in the cases of partial observation procedures for computing weak suboptimal strategies are suggested and these are computed for a variety of examples

an explicit fourth order staggered finite difference time domain method for maxwell s equations a new explicit fourth order accurate staggered finite difference time domain fdtd scheme is proposed and applied to electromagnetic wave problems it is fourth order accurate in both space and time conditionally stable and highly efficient with respect to yee s scheme and still retains much of the original simplicity of yee s scheme both extension to perfectly matched layers and modification to deal with dielectric interfaces and perfectly conducting boundaries of the scheme have also been presented numerical examples are shown to illustrate the efficiency of the method

distance for degree raising and reduction of triangular bezier surfaces the problem of degree reduction and degree raising of triangular bezier surfaces is considered the l 2 and l 2 measures of distance combined with the least squares method are used to get a formula for the bezier points the methods use the matrix representations of the degree reduction and degree raising

comparative study between two navier stokes algorithms for transonicairfoils a critical examination of several aspects of the numerical simulation of high reynolds number transonic airfoil flows is presented subcritical and supercritical flowfields about an aft cambered airfoil were generated by solving the mass averaged navier stokes equations with turbulence incorporated through an algebraic eddy viscosity model the governing equations were solved on curvilinear body fitted grids utilizing two different algorithms i e maccormack s explicit and beam warming implicit the numerical uncertainties associated with different schemes grid resolution artificial viscosity and far field boundary placement were investigated and found to be of the same order of magnitude or less than the corresponding uncertainties in the available experimental data comparison of computed and experimental results showed good prediction of all the essential flow features however detailed comparison of velocity profiles pointed out deficiencies of the turbulence model downstream of the shock wave boundary layer interaction in the airfoil cove and in the near wake thinlayer and navier stokes computed results were found to be in excellent agreement with each other

the weighted dual functionals for the univariate bernstein basis abstract we find an explicit formula for the weighted dual functions of the bernstein polynomials with respect to the jacobi weight function using the usual inner product in the hilbert space l 2 0 1 we define the weighted dual functionals of the bernstein polynomials which are used to find the coefficients in the least squares approximation

a multistage reduction technique for feedback stabilizing distributed time lag systems abstract a direct pole relocation theory is advanced for linear time invariant systems with distributed delays in both state and control variables the principal tools of the theory include i the finite cardinality of the unstable spectrum ii a set of matrices each of which is a left zero of the system characteristic quasi polynomial matrix and iii a linear transformation which reduces the delay system to a sufficiently high order delay free system whose spectrum contains the delay system unstable spectrum it is shown that if the delay system is spectrally stabilizable then it shares a common feedback stabilizing control law with its delay free counterpart this point of contact with a delay free system permits the determination of the control law using well established ordinary system methods the workability of the approach hinges on the ability to partition the unstable spectrum augmented with additional poles from the stable spectrum if necessary into n symmetric sets when this partition is impossible a spectral controllability invariance theorem facilitates resolution of the problem

two dimensional elliptic grid solver using boundary grid control and curvature correction the thompson method of explicitly evaluating control function of the elliptic grid solver is employed to improve local grid quality around convex and concave boundaries the curvature correction is principally added to the grid lines around a boundary and is quickly attenuated as the grid points move inward the algebraic advancing front method is employed to estimate two levels of orthogonal grids for evaluating the control functions at the boundary if the grid equation based on the cauchy riemann relation with and as independent variables is employed numerical examinations show that a lengthy trial and error procedure is required to get a satisfactory grid distribution in contrast the method based on the cauchy riemann relation with x and y as independent variables which benefits from the maximum principle not only removes the undesired grid clustering and diluting easily but also effectively improves overall grid smoothness and slightly enhances boundary grid orthogonality

optimal multi degree reduction of triangular bezier surfaces with corners continuity in the norm l2 this paper derives an approximation algorithm for multi degree reduction of a degree n triangular bezier surface with corners continuity in the norm l 2 the new idea is to use orthonormality of triangular jacobi polynomials and the transformation relationship between bivariate jacobi and bernstein polynomials this algorithm has a very simple and explicit expression in matrix form i e the reduced matrix depends only on the degrees of the surfaces before and after degree reduction and the approximation error of this degree reduced surface is minimum and can get a precise expression before processing of degree reduction combined with surface subdivision the piecewise degree reduced patches possess global c 0 continuity finally several numerical examples are presented to validate the effectiveness of this algorithm

an explicit fourth order orthogonal curvilinear staggered grid fdtd method for maxwell s equations the explicit fourth order staggered finite difference time domain scheme previously proposed for a cartesian grid is extended to maxwell s equations in an orthogonal curvilinear coordinate system and applied to electromagnetic wave problems a simple technique is also presented for generating orthogonal curvilinear grids that conform to the material boundaries and interfaces of the problem numerical experiments are presented to illustrate the efficiency and accuracy of the method

on generation of orthogonal grids this study concerns some problems of generation of orthogonal grids the use of scaled laplace equations is analyzed and the problems related to their application are discussed the issues of existence and orthogonality of mappings are studied and some analytical examples showing inconsistency of scaled laplace equations with orthogonality requirements are constructed finally a simple algorithm for generating the orthogonal mappings based on decoupled form of equations is presented

stochastic projection schemes for deterministic linear elliptic partial differential equations on random domains we present stochastic projection schemes for approximating the solution of a class of deterministic linear elliptic partial differential equations defined on random domains the key idea is to carry out spatial discretization using a combination of finite element methods and stochastic mesh representations we prove a result to establish the conditions that the input uncertainty model must satisfy to ensure the validity of the stochastic mesh representation and hence the well posedness of the problem finite element spatial discretization of the governing equations using a stochastic mesh representation results in a linear random algebraic system of equations in a polynomial chaos basis whose coefficients of expansion can be non intrusively computed either at the element or the global level the resulting randomly parametrized algebraic equations are solved using stochastic projection schemes to approximate the response statistics the proposed approach is demonstrated for modeling diffusion in a square domain with a rough wall and heat transfer analysis of a three dimensional gas turbine blade model with uncertainty in the cooling core geometry the numerical results are compared against monte carlo simulations and it is shown that the proposed approach provides high quality approximations for the first two statistical moments at modest computational effort

generation of orthogonal grids with control of spacing abstract a methodology for generating orthogonal curvilinear grids is applied to two dimensional domains an important feature of the methodology is its ability to control effectively the grid spacing especially near the boundaries this paper summarizes the governing equations used for grid generation then the numerical procedure is described with special emphasis on the scheme used to enhance stability and accuracy of the solution the significance of the distortion function and the way it is used to control grid spacing are illustrated in geometries commonly found in engine combustion chambers the influence of various parameters including number of grid points relaxation factor and range of values of the distorion function on the performance of the method are also investigated it is concluded that the methodology can successfully produce smooth orthogonal grids with control of spacing in symmetric and nonsymmetric domains

a note on constrained degree reduction of polynomials in bernstein bezier form over simplex domain in the paper h s kim y j ahn constrained degree reduction of polynomials in bernstein bezier form over simplex domain j comput appl math 216 2008 14 19 kim and ahn proved that the best constrained degree reduction of a polynomial over d dimensional simplex domain in l 2 norm equals the best approximation of weighted euclidean norm of the bernstein bezier coefficients of the given polynomial in this paper we presented a counterexample to show that the approximating polynomial of lower degree to a polynomial is virtually non existent when d 2 furthermore we provide an assumption to guarantee the existence of solution for the constrained degree reduction

bezier representation of the constrained dual bernstein polynomials abstract explicit formulae for the bezier coefficients of the constrained dual bernstein basis polynomials are derived in terms of the hahn orthogonal polynomials using difference properties of the latter polynomials efficient recursive scheme is obtained to compute these coefficients applications of this result to some problems of cagd is discussed

robust multivariable pi controller for infinite dimensional systems a robust multivariable controller is introduced for a class of distributed parameter systems the system to be controlled is given as dot x ax bu y cx in a banach space the purpose of the control which is based on the measurement y is to stabilize and regulate the system so that y t rightarrow y r as t rightarrow infty where y r is a constant reference vector under the assumptions that operator a generates a holomorphic stable semigroup b is linear and bounded c is linear and a bounded and the input and output spaces are of the same dimension a necessary and sufficient condition is found for the existence of a robust multivariable controller this controller appears to be a multivariable pi controller also a simple necessary criterion for the existence of a decentralized controller is derived the tuning of the controller is discussed and it is shown that the i part of the controller can be tuned on the basis of step responses without exact knowledge of the system s parameters the presented theory is then used as an example to control the temperature profile of a bar with the dirichlet boundary conditions

application of bubnov galerkin formulation to orthogonal grid generation abstract a bubnov galerkin formulation is used to solve an elliptic grid generation system by using linear and quadratic isoparametric elements good orthogonality characteristics are obtained for symmetric and non symmetric physical domains using both complete boundary correspondence or a combination of dirichlet and neuman boundary conditions the method exhibits excellent stability and requires a low number of iterations to attain convergence results are compared with those presented in e d chikliwala and y c yortsos j comput phys 57 391 1985

feedback stabilization of linear autonomous time lag systems a stabilization theory which employs well established finite dimensional control system tools is developed for the stabilization of linear autonomous time lag systems the main ideas include 1 a set whose elements are matrices each of which is a left zero of the system characteristic quasi polynomial matrix and 2 a linear transformation which reduces the delay system to a delay free system whose state matrix is a direct sum of n elements of the matrix set where n is some positive integer from the definition of this matrix set it is shown that each of its elements inherits its spectrum from that of the delay system so that by design the system unstable poles may be embedded in the spectrum of the delay free system under the assumption of spectral stabilizability it is then shown how to obtain a stabilizing feedback control law on the basis of the delay free system numerical examples are presented to confirm the theory

orthogonal mapping in two dimensions abstract a method for the generation of orthogonal boundary fitted curvilinear coordinates for arbitrary simply and doubly connected domains is developed on the basis of the theory of quasi conformal mappings of quadrilaterals and of previous work by ryskin and leal the method has useful applications in orthogonal grid generation in two dimensional and axi symmetric domains and in the extension of rapid elliptic solvers and spectral methods to complex geometries a new technique for the calculation of the conformal module of quadrilaterals is also presented

generation of curvilinear coordinates on multiply connected regions with boundary singularities a new algorithm to generate smooth two dimensional boundary conforming coordinates with grid lines control on multiply connected regions including boundary singularities is devised the technique fits into the category of elliptic grid generators since it is based on the numerical solution of poisson equations the physical domain d is transformed into a topologically equivalent connected rectangular domain d by defining a branch cut inside d control of the grid line spacing over the multiply connected regions is established by appropriately distributing nodal points on the branch cut and on other boundary curves simple and computationally convenient expressions for the control functions present in the poisson system are obtained from this initial distribution of points thus a natural link between clustering properties and the control functions is created fixing the nodal points on the branch cut causes the convergence of the numerical method about the singular boundary points an iterative smoothing process relocating the branch cut is described as a result smooth grids are obtained grid quality analysis shows that coordinate line orthogonality in the new meshes is superior than in similar grids obtained from a well known commercial software the algorithm is successfully tested over a variety of multiply connected domains the computational advantage of this novel grids is revealed by using them to numerically model the vibrations of complexly shaped annular membranes

modified jacobi bernstein basis transformation and its application to multi degree reduction of bezier curves this paper reports new modified jacobi polynomials mjps we derive the basis transformation between mjps and bernstein polynomials and vice versa this transformation is merging the perfect least square performance of the new polynomials together with the geometrical insight of bernstein polynomials the mjps with indexes corresponding to the number of endpoints constraints are the natural basis functions for least square approximation of bezier curves using mjps leads us to deal with the constrained jacobi polynomials and the unconstrained jacobi polynomials as orthogonal polynomials the mjps are automatically satisfying the homogeneous boundary conditions thereby the main advantage of using mjps in multi degree reduction of bezier curves on computer aided geometric design cagd is that the constraints in cagd are also satisfied and that decreases the steps of multi degree reduction algorithm several numerical results for the multi degree reduction of bezier curves on cagd are given we establish and prove the basis transformation between the modified jacobi polynomials mjps and bernstein polynomials and vice versa this transformation is merging the perfect least square performance of the mjps with the geometrical insight of the bernstein form the mjps with indexes corresponding to the number of endpoint constraints are the natural basis functions for least square approximation of bezier curves several numerical results are considered

weighted dual functions for bernstein basis satisfying boundary constraints in this paper we consider the issue of dual functions for the bernstein basis which satisfy boundary conditions the jacobi weight function with the usual inner product in the hilbert space are used some examples of the transformation matrices are given some figures for the weighted dual functions of the bernstein basis with respect to the jacobi weight function satisfying boundary conditions are plotted we discuss special cases of the jacobi weight function as the legendre weight function and the chebyshev weight functions of the first second and third kinds

the jacobi last multiplier for linear partial difference equations we present a discretization of the jacobi last multiplier with some applications to the computation of solutions of linear partial difference equations

construction of dual bases let b n b 0 b 1 b n n 0 1 n n n be the sets of linearly independent functions we give a simple method of construction the dual functions d n d 0 n d 1 n d n n 0 n n satisfying the following conditions spand n spanb n and d i j 0 i j n n where d i i 1 d i j 0 for i j and is a given inner product the proposed algorithm allows us to construct all the sets of the dual functions d 0 d 1 d n in the time o n 3 where n is a natural number four illustrative examples presenting the possible applications of obtained results are given

adaptive suboptimal control of a linear system with bounded disturbance abstract an adaptive suboptimal control of a linear discrete system with unknown parameters is proposed an additive disturbance v t acting on the system is supposed to be uniformly bounded the criterion is sup v t i y 1 u 1 where y t is the output u t is the control the adaptive control law gives almost the same guaranteed value of the criterion as the optimal linear feedback does for a system with known parameters

a scheme for interpolation with trigonometric spline curves we present a method for the interpolation of a given sequence of data points with c n continuous trigonometric spline curves of order n 1 n 1 that are produced by blending elliptical arcs ready to use explicit formulae for the control points of the interpolating arcs are also provided each interpolating arc depends on a global parameter 0 that can be used for global shape modification associating non negative weights with data points rational trigonometric interpolating spline curves can be obtained where weights can be used for local shape modification the proposed interpolation scheme is a generalization of the overhauser spline and it includes a c n bezier spline interpolation method as the limiting case 0

an iterative algorithm for g 2 multiwise merging of bezier curves this paper presents an iterative algorithm for g 2 multiwise merging of bezier curves by using the g 2 constraint the l 2 distance is represented after simplification as a quartic polynomial in two parameters relating to the magnitudes of end tangents of the merged curve these two parameters are restricted in a feasible region in order for the merged curve to preserve the specified directions of end tangents then g 2 multiwise merging is formulated as a constrained minimization problem and the classic projected newton method is applied to find the minimizer some extensions of multiwise merging using g 3 constraints other energy functionals and curve representations are also outlined several comparative examples are provided to demonstrate the effectiveness of the proposed method

an approximation method to circular arcs abstract modern cad cam systems are not able to dispose the circles represented by the implicit equations or the parametric equations with trigonometric function parametric rational polynomials are used to approximate the circular arcs a new approximation method by means of quartic bezier curves is presented to effectively approximate the circular arcs it is shown that our method provides the approximation of order eight and yields a smaller error than previous methods by equally spaced subdivision of circular arcs the curvature continuous spline approximation algorithm of the circular arcs is given

three level incentive strategies in differential games the three level incentive strategy in a non linear differential game ia defined the separability in obtaining incentive strategies is shown the sufficient condition for a three level incentive strategy in a linear quadratic differential game is derived it is shown that a gradient method is applicable for finding a three level incentive strategy in a linear quadratic differential game

separation control by alternating tangential blowing suction at multiple slots a method of separation control using alternating tangential blowing suction at small speeds on multiple slots was proposed and the properties of this method were studied by applying it to the flow control of a thick airfoil the method of numerically solving the reynolds averaged navier stokes equations was employed for the study using alternating tangential blowing suction with small speeds in the blowing phase the boundary layer velocity profiles downstream of the slot are made fuller and more separation resistant and in the suction phase the boundary layer velocity profiles both up and downstream of the slot are made fuller and more separation resistant for the airfoil considered in the paper which is of 40 percent thickness and has ten slots with a peak velocity of about 1 5

construction of dual b spline functions using new properties of dual functions we show that the method of constructing the dual bases recently presented in p wozny construction of dual bases journal of computational and applied mathematics 245 2013 75 85 can be significantly simplified next we propose a simple algorithm of constructing the dual b spline functions which can be useful in solving some approximation problems related to numerical analysis or computer graphics

optimal bases for a class of mixed spaces and their associated spline spaces we consider spaces of the form span where the functions u i i 1 4 are algebraic polynomials or trigonometric or hyperbolic functions we find intervals 0 a where we can guarantee that the spaces possess normalized totally positive bases and so shape preserving representations we construct their normalized b bases with optimal shape preserving properties we also present a unified approach to deal with the associated spline spaces a recursive procedure to obtain bases with optimal shape preserving and stability properties is presented

transfinite mappings and their application to grid generation the two essential ingredients of any boundary value problem are the field equations which describe the physics of the problem and a set of relations which specify the geometry of the problem domain mesh generators or grid generators are preprocessors whicd decompose the problem domain into a large number of interconnected finite elements or curvilinear finite difference stencils a number of such techiques have been developed over the past decade to alleviate the frustation and reduce the time involved in the tedious manual subdividing of a complex shaped region or 3 d structure into finite elements our purpose here is to describe how the techniques of bivariate and trivariate blending function interpolation which were originally developed for and applied to geometric problems of computer aided design of sculptured surface and 3 d solids can be adapted and applied to the geometric problems of grid generation in contrast to other techniques which require the numerical solution of complex partial differential equations and hence a great deal of computing the transfinite methods proposed herein are computationally inexpensive

gram matrix of bernstein basis this note presents explicit expressions for the inverses of the gram matrix of the bernstein basis and its principal submatrices by taking the advantages of the transformations between the bernstein basis and the constrained dual bernstein basis using the symmetry property fast calculation of these matrices and their inverses is achieved significant improvements are obtained for applications including polynomial approximation of functions and degree reduction of bezier curves

suboptimal control of linear discrete stochastic systems with linear input constraints optimal control of linear discrete stochastic systems with linear input constraints is considered a lower bound on the achievable minimum of the loss function is given a suboptimal control strategy is derived by using a truncated taylor series to approximate the expected future loss in the bellman equation the performance of the suboptimal controller is studied using monte carlo simulation and the obtained loss is compared with the lower bound

explicit algorithms for multiwise merging of bezier curves this paper presents a novel scheme called c r s multiwise merging for merging multiple segments of bezier curves using a single bezier curve it is considered as an extension of the existing pairwise merging to avoid the limitations caused by recursively applying pairwise merging to the multiple case an explicit algorithm is developed to obtain the merged curve which preserves c r and c s continuity at the endpoints and is optimal in the sense that the l 2 or l 2 distance is minimized as an application we develop explicit algorithms for g 1 multiwise merging always producing better results than c 1 multiwise merging

suboptimal control of linear tracking systems with time delays the optimal control solution of a dynamical system with time delays is generally very difficult to obtain in this paper a uniformly convergent iterative algorithm is developed to obtain a partially closed loop suboptimal control function for the time varying linear tracking system with multiple time delays in both state variables and control variables such that a proposed quadratic cost functional was minimized the essential feature of this method was to treat the time delayed functions evaluated from the k 1 th iteration step as perturbations to the kth optimizing iteration of the non delayed system the convergency rate of suboptimal solution was very fast

approximating tensor product bezier surfaces with tangent plane continuity we present a simple method for degree reduction of tensor product bezier surfaces with tangent plane continuity in l 2 norm continuity constraints at the four corners of surfaces are considered so that the boundary curves preserve endpoints continuity of any order a we obtain matrix representations for the control points of the degree reduced surfaces by the least squares method a simple optimization scheme that minimizes the perturbations of some related control points is proposed and the surface patches after adjustment are c continuous in the interior and g 1 continuous at the common boundaries we show that this scheme is applicable to surface patches defined on chessboard like domains

new boundary constraints for elliptic systems used in grid generation problems new boundary constraints for elliptic partial differential equations as used in grid generation problems in generalized curvilinear coordinate systems are proposed in this paper these constraints based on the principle of local conservation of thermal energy in the vicinity of the boundaries are derived using the green s theorem they uniquely determine the so called decay parameters in the inhomogeneous terms of these elliptic systems these constraints are designed for boundary clustered grids where large gradients in physical quantities need to be resolved adequately it is observed that the present formulation also works satisfactorily for mild clustering therefore a closure for the decay parameter specification for elliptic grid generation problems has been provided resulting in a fully automated elliptic grid generation technique thus there is no need for a parametric study of these decay parameters since the new constraints fix them uniquely it is also shown that for neumann type boundary conditions these boundary constraints uniquely determine the solution to the internal elliptic problem thus eliminating the nonuniqueness of the solution of an internal boundary value grid generation problem with neumann boundary conditions

constrained multi degree reduction of triangular bezier surfaces using dual bernstein polynomials we propose a novel approach to the problem of multi degree reduction of bezier triangular patches with prescribed boundary control points we observe that the solution can be given in terms of bivariate dual discrete bernstein polynomials the algorithm is very efficient thanks to using the recursive properties of these polynomials the complexity of the method is o n 2m 2 n and m being the degrees of the input and output bezier surfaces respectively if the approximation with appropriate boundary constraints is performed for each patch of several smoothly joined triangular bezier surfaces the result is a composite surface of global c r continuity with a prescribed order r some illustrative examples are given

regulator problem for infinite dimensional systems abstract in this paper a regulator problem is investigated for an infinite dimensional system with constant disturbances the regular problem is to determine a feedback control law which stabilizes and regulates the system a design procedure of a regulator is proposed which can be realized in finite dimensional theories and techniques for an infinite dimensional system

generation of orthogonal and nearly orthogonal coordinates with gridcontrol near boundaries a numerical procedure is presented for the generation of boundary fitted curvilinear coordinates with controllable mesh spacing and orthogonality the technique is based on a straightforward two step procedure involving the solution of poisson s equation the method is capable of generating orthogonal grids with partial control of the mesh spacing or nearly orthogonal grids with strict control of the mesh spacing the inhomogeneous terms in the poisson equations are automatically adjusted during the grid generation the technique is applicable to two dimensional simply connected regions typical of airfoils cascades diffusers and inlets

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

unified and extended form of three types of splines the three types refer to polynomial trigonometric and hyperbolic splines in this paper we unify and extend them by a new kind of spline ue spline for short defined over the space cos wt sin wt 1 t t l where l is an arbitrary nonnegative integer w is a frequency sequence w i a i a i r existing splines such as usual polynomial b splines cb splines hb splines nuat splines ah splines fb splines and the third form fb splines etc are all special cases of ue splines ue splines inherit most properties of usual polynomial b splines and enjoy some other advantageous properties for modelling they can exactly represent classical conics the catenary the helix and even the eight curve a kind of snake like curves etc

degree reduction of composite bezier curves this paper deals with the problem of multi degree reduction of a composite bezier curve with the parametric continuity constraints at the endpoints of the segments we present a novel method which is based on the idea of using constrained dual bernstein polynomials to compute the control points of the reduced composite curve in contrast to other methods ours minimizes the l2 error for the whole composite curve instead of minimizing the l2 errors for each segment separately as a result an additional optimization is possible examples show that the new method gives much better results than multiple application of the degree reduction of a single bezier curve

composite three dimensional grids generated by elliptic systems a technique is presented for constructing a boundary conforming grid throughout a general threedimensional flow region as a composite of subregion grids each subregion grid is generated numerically by solving a nonlinear system of elliptic equations the boundary values represent nodal points in a quasi twodimensional grid that covers the curved surface bounding the subregion and are generated numerically by a modified elliptic system a primary feature of the technique is that the composite three dimensional grid remains both continuous and smooth across the surface of juncture between any two adjoining subregions sample results are given for a wing body combination

three dimensional elliptic grid generation with fully automatic boundary constraints a new procedure for generating smooth uniformly clustered three dimensional structured elliptic grids is presented here which formulates three dimensional boundary constraints by extending the two dimensional counterpart presented by the author earlier this fully automatic procedure obviates the need for manual specification of decay parameters over the six bounding surfaces of a given volume grid the procedure has been demonstrated here for the mars science laboratory msl geometries such as aeroshell and canopy as well as the inflatable aerodynamic decelerator iad geometry and a 3d analytically defined geometry the new procedure also enables generation of single block grids for such geometries because the automatic boundary constraints permit the decay parameters to evolve as part of the solution to the elliptic grid system of equations these decay parameters are no longer just constants as specified in the conventional approach but functions of generalized coordinate variables over a given bounding surface since these decay functions vary over a given boundary orthogonal grids around any arbitrary simply connected boundary can be clustered automatically without having to break up the boundaries and the corresponding interior or exterior domains into various blocks for grid generation the new boundary constraints are not limited to the simply connected regions only but can also be formulated around multiply connected and isolated regions in the interior the proposed method is superior to other methods of grid generation such as algebraic and hyperbolic techniques in that the grids obtained here are c 2 continuous whereas simple elliptic smoothing of algebraic or hyperbolic grids to enforce c 2 continuity destroys the grid clustering near the boundaries

explicit g 1 approximation of conic sections using bezier curves of arbitrary degree in this paper we propose an explicit and effective method for g 1 approximation of conic sections with arbitrary degree polynomial curves the geometric constraints can be expressed by two variables then we construct an objective function as the alternative approximation error function in the l 2 norm which is a quadratic function in these two variables to minimize the objective function is equivalent to solving a system of linear equations with two variables since its coefficient matrix is invertible we can explicitly obtain the unique solution and then derive the explicit g 1 approximation of conic sections by bezier curves of arbitrary degree the proposed method has an optimal approximation in the l 2 norm i e the l 2 distance error reaches minimum and also can process the conic section with center angle larger than without subdivision scheme finally numerical examples demonstrate the effectiveness of our method

2d orthogonal grid generation with boundary point distribution control a numerical method for 2d orthogonal grid generation with control of the boundary point distribution is presented the method is based on the solution of a system of partial differential equations the grid cell aspect ratio the so called distortion function is calculated from its definition equation in the entire domain the method allows the specification of the boundary point distribution in all the boundaries and also the distance of the first grid node to the boundary the method is successfully applied to several geometries including geometries with grid singularities and grids around airfoils and cross sections of ship sterns

best polynomial degree reduction on q lattices with applications to q orthogonal polynomials we study the best polynomial degree reduction with respect to the q l2 norm we study a finite analogue with respect to finite q lattices we present applications to q orthogonal polynomials we show that a weighted least squares approximation of q bezier coefficients provides the best polynomial degree reduction in the q l2 norm we also provide a finite analogue of this result with respect to finite q lattices and we present applications of these results to q orthogonal polynomials

algorithms for bound constrained quadratic programming problems we present an algorithm which combines standard active set strategies with the gradient projection method for the solution of quadratic programming problems subject to bounds we show in particular that if the quadratic is bounded below on the feasible set then termination occurs at a stationary point in a finite number of iterations moreover if all stationary points are nondegenerate termination occurs at a local minimizer a numerical comparison of the algorithm based on the gradient projection algorithm with a standard active set strategy shows that on mildly degenerate problems the gradient projection algorithm requires considerable less iterations and time than the active set strategy on nondegenerate problems the number of iterations typically decreases by at least a factor of 10 for strongly degenerate problems the performance of the gradient projection algorithm deteriorates but it still performs better than the active set method

suboptimal control of discrete stochastic amplitude constrained systems abstract optimal control of linear discrete time stochastic systems with an amplitude constraint on the input is considered a suboptimal controller is obtained by considering linear feedback laws with saturation the variances of the closed loop system are evaluated by approximating the stationary probability distribution of the state with a gaussian distribution the gradient of the loss with respect to the feedback gain is evaluated and an iterative algorithm to optimize the regulator parameters is given numerical examples are used to evaluate the control performance of the suboptimal strategy the obtained loss is compared with a lower bound on the minimum of the loss which is achievable in amplitude constrained control the results show that the suboptimal strategy may give near optimal control performance

a suboptimal terminal controller for linear discrete time systems abstract in this work we consider what happens when the receding horizon control laws are designed with finite final state penalization matrices a sufficient condition in terms of system and performance index parameters is given for the discrete receding horizon control rhc laws to be stabilizing when used for suboptimal regulation this condition enables the weighting matrices in the quadratic criterion to be pro assigned so that stability of design is secured the exact value of rhc cost in terms of optimal cost and an easily computable upper bound on its deviation from the optimal cost is found fast algorithms are included to calculate the particular rhc feedback gains introduced dual results in estimation theory are pointed out in this way rhc is proposed as an attractive suboptimal alternative to the optimal terminal control of linear discrete time systems

nearly orthogonal two dimensional grid generation with aspect ratio control an improved method for nearly orthogonal grid generation is presented in this study the generating system is based on solution of a system of partial differential equations with finite difference discretization to prevent grid lines from collapsing onto each other the grid cell aspect ratio is controlled by functions that limit excessive ratios bounding all the aspect ratios is essential for high quality numerical approximations using such grid based methods as finite elements finite differences or finite volumes the influence of the number of grid points type of boundary and intensity of the grid quality control function and grid properties are investigated specification of both boundary point distribution on all sides and moving boundaries is used the proposed method is applied to various test problems from the literature this method is shown to provide a good balance between controlling grid orthogonality and cell aspect ratio c 2001 academic press

a discontinuity capturing crosswind dissipation for the finite element solution of the convection diffusion equation abstract to avoid the local oscillations that still remain using the streamline upwind petrov galerkin formulation for the scalar convection diffusion equation the introduction of a nonlinear crosswind dissipation is proposed it is shown that the method is less overdiffusive than other discontinuity capturing techniques and has better numerical behavior the design of the crosswind diffusion is based on the study of the discrete maximum principle for some simple cases

discretization of free surface flows and other moving boundary problems a system of elliptic partial differential equations and boundary conditions has been developed for generating boundary fitted element discretizations of two dimensional free and moving boundary problems terms in the differential equations are scaled for dimensional homogeneity and adjustable weighting of orthogonality smoothness and concentration of the coordinate mesh they govern grid points become finite element nodes mapped isoparametrically or subparametrically from a simple or patched computational domain concentration terms contain control functions and parameters that influence node spacing along each coordinate independently overall control is by patchwise parameters and functions successful selection of these to follow deforming flow regions is straightforward and is illustrated by analysis of steady and transient slide coating flows

multi degree reduction of tensor product bezier surfaces with general boundary constraints abstract we propose an efficient approach to the problem of multi degree reduction of rectangular bezier patches with prescribed boundary control points we observe that the solution can be given in terms of constrained bivariate dual bernstein polynomials the complexity of the method is o mn 1 n 2 with m min m1 m2 where n1 n2 and m1 m2 is the degree of the input and output bezier surface respectively if the approximation with appropriate boundary constraints is performed for each patch of several smoothly joined rectangular bezier surfaces the result is a composite surface of global cr continuity with a prescribed r 0 in the detailed discussion we restrict ourselves to r 0 1 which is the most important case in practical application some illustrative examples are given

a pi controller for distributed delay systems abstract a finite dimensional control theory is developed for the design of a proportional plus integral control law for tracking step command inputs in general autonomous time lag systems with distributed heredity in state control and output variables with respect to an ordinary differential equation describing the delay system unstable modes an auxiliary tracking problem is posed this auxiliary problem derives its importance from the fact that its solution directly yields a solution to the original tracking problem this point of contact with a finite dimensional system permits the application of well tested ordinary systems tracking theory to the tracking problem in time lag systems

on one dimensional stretching functions for finite difference calculations abstract the class of one dimensional stretching functions used in finite difference calculations is studied for solutions containing a highly localized region of rapid variation simple criteria for a stretching function are derived using a truncation error analysis these criteria are used to investigate two types of stretching functions one is an interior stretching function for which the location and slope of an interior clustering region are specified the simplest such function satisfying the criteria is found to be one based on the inverse hyperbolic sine it was first employed by thomas et al aiaa j 10 1972 887 the other type of function is a two sided stretching function for which the arbitrary slopes at the two ends of the one dimensional interval are specified the simplest such general function is found to be one based on the inverse tangent the special case where the slopes were both equal and greater than one was first employed by roberts the general two sided function has many applications in the construction of finite difference grids examples of such applications are found in the listed references

construction of partial difference schemes i the clairaut schwarz young theorem on the lattice difference operators in several variables do not always commute their commutation depends on the lattice equations in this paper we study the conditions over a lattice to allow commutativity of these difference operators and their implications for the discretization of partial differential equations