fourier transform of a polygonal shape function and its application in electromagnetics the shape function s x y of a planar aperture sigma is defined ashaving value one inside sigma and zero outside the two dimensional fourier transform of s x y when sigma is an n sided polygon is considered and a closed form exact solution is presented applications of these results are described for 1 the calculation of the secondary pattern of a reflector antenna with a noncircular aperture and 2 high frequency transmission through a periodic frequency selective surface

the mollification method and the numerical solution of the inverse heat conduction problem by finite differences abstract the inverse heat conduction problem involves the calculation of surface heat flux and or temperature histories from transient measured temperatures inside solids we consider the one dimensional semi infinite linear case and present a new solution algorithm based on a data filtering interpretation of the mollification method that automatically determines the radius of mollification depending on the amount of noise in the data and finite differences a fully explicit and stable space marching scheme is developed we describe several numerical experiments of interest showing that the new procedure is accurate and stable with respect to perturbations in the data even for small dimensionless time steps

a penalized neumann control approach for solving an optimal dirichlet control problem for the navier stokes equations we introduce a penalized neumann boundary control approach for solving an optimal dirichlet boundary control problem associated with the two or three dimensional steady state navier stokes equations we prove the convergence of the solutions of the penalized neumann control problem the suboptimality of the limit and the optimality of the limit under further restrictions on the data we describe the numerical algorithm for solving the penalized neumann control problem and report some numerical results

a mollified space marching finite differences algorithm for the inverse heat conduction problem with slab symmetry abstract the one dimensional inverse heat conduction problem ihcp for a slab is considered a new solution algorithm based on a data filtering interpretation of the mollification method is presented and a fully explicit space marching finite differences scheme is developed after showing the numerical stability of the algorithm the efficiency of the method is demonstrated by means of several examples the numerical results show the procedure to be very useful in the presence of noisy data even for some nonlinear cases

reduced basis approximation of parametrized optimal flow control problems for the stokes equations this paper extends the reduced basis method for the solution of parametrized optimal control problems presented in negri et al 2013 to the case of noncoercive elliptic equations such as the stokes equations we discuss both the theoretical properties with particular emphasis on the stability of the resulting double nested saddle point problems and on aggregated error estimates and the computational aspects of the method then we apply it to solve a benchmark vorticity minimization problem for a parametrized bluff body immersed in a two or a three dimensional flow through boundary control demonstrating the effectivity of the methodology

determining surface temperatures from interior observations we consider the inverse heat transfer problem for the case of the one dimensional heat equation in a quarter plane and present a new solution algorithm together with error bounds of logarithmic convexity type our method is based on regularizing the initial value problem for the heat equation run sideways we describe a numerical experiment with rapidly varying temperature profiles and explain how the method can be extended to problems with temperature dependent thermal properties

constrained dirichlet boundary control in l 2 for a class of evolution equations optimal dirichlet boundary control based on the very weak solution of a parabolic state equation is analyzed this approach allows us to consider the boundary controls in l 2 which has advantages over approaches which consider control in sobolev spaces involving fractional derivatives pointwise constraints on the boundary are incorporated by the primal dual active set strategy its global and local superlinear convergences are shown a discretization based on space time finite elements is proposed and numerical examples are included

compact range performance evaluation using aperture near field angular spectrums for a compact range cr the near field angular spectrum can be obtained by making an inverse fourier transform of its aperture spatial near field the characteristics of the aperture design have been carried by the aperture near field angular spectrum the direct wave and edge diffracted waves from the aperture can be separated and distinguished clearly in the angular domain therefore the cr performance can be evaluated and improved accordingly in this paper based on the aperture field convolution method and plane wave spectrum pws theory near field angular spectrums for different apertures have been computed the general criteria for the cr aperture design have been discussed the selections of the whole aperture shape the serration height the serration base length the serration number and the serration shape including isosceles and right angled triangles have been compared at last the near fields of an optimized serrated edge aperture and an original circular aperture in different transverse planes have been computed and compared the width and height of the apertures are all 30 wavelengths in free space in order to suppress the sidelobes a chebyshev window with 50 db sidelobes has been adopted in the spatial angular transform

approximations of very weak solutions to boundary value problems standard weak solutions to the poisson problem on a bounded domain have square integrable derivatives which limits the admissible regularity of inhomogeneous data the concept of solution may be further weakened in order to define solutions when data is rough such as for inhomogeneous dirichlet data that is only square integrable over the boundary such very weak solutions satisfy a nonstandard variational form u v g v a galerkin approximation combined with an approximation of the right hand side g defines a finite element approximation of the very weak solution applying conforming linear elements leads to a discrete solution equivalent to the text book finite element solutionto the poisson problem in which the boundary data is approximated by l2 projections the l2 convergence rate of the discrete solution is o hs for some s in 0 1 2 that depends on the shape of the domain assuming a polygonal two dimensional or polyhedral three dimensional domain without slits and only square integr

an energy space finite element approach for elliptic dirichlet boundary control problems in this paper we present a finite element analysis for a dirichlet boundary control problem where the dirichlet control is considered in a convex closed subspace of the energy space h 1 2 gamma h 1 2 as an equivalent norm in h 1 2 gamma h 1 2 we use the energy norm induced by the so called steklov poincare operator which realizes the dirichlet to neumann map and which can be implemented by using standard finite element methods the presented stability and error analysis of the discretization of the resulting variational inequality is based on the mapping properties of the solution operators related to the primal and adjoint boundary value problems and their finite element approximations some numerical results are given which confirm on one hand the theoretical estimates but on the other hand indicate the differences when modelling the control in l 2 gamma l 2

the space marching solution of the inverse heat conduction problem and the identification of the initial temperature distribution abstract it is shown that the usual data for the inverse heat conduction problem ihcp uniquely defines the initial condition for the direct problem in a one dimensional finite slab a suitable model problem is also analyzed for the semi infinite conductor case a space marching method is implemented to numerically solve both the ihcp and the identification of the initial temperature distribution several examples of interest are presented

an efficient hybrid go pws algorithm to analyze conformal serrated edge reflectors for millimeter wave compact range a method to analyze parabolic reflectors with arbitrary piecewise rim is presented in this communication this kind of reflectors when operating as collimators in compact range facilities needs to be large in terms of wavelength their analysis is very inefficient when it is carried out with fullwave mom techniques and it is not very appropriate for designing with po techniques also fast go formulations do not offer enough accuracy to reach performance results the proposed algorithm is based on a go pws hybrid scheme using analytical as well as non analytical formulations on one side an analytical treatment of the polygonal rim reflectors is carried out on the other side non analytical calculi are based on efficient operations such as m2 order 2 dimensional fft a combination of these two techniques in the algorithm ensures real ad hoc design capabilities reached through analysis speedup the purpose of the algorithm is to obtain an optimal conformal serrated edge reflector design through the analysis of the field quality within the quiet zone that it is able to generate in its forward half space

boundary element methods for variational inequalities in this paper we present a priori error estimates for the galerkin solution of variational inequalities which are formulated in fractional sobolev trace spaces i e in widetilde h 1 2 gamma h 1 2 in addition to error estimates in the energy norm we also provide by applying the aubin nitsche trick for variational inequalities error estimates in lower order sobolev spaces including l 2 gamma l 2 the resulting discrete variational inequality is solved by using a semi smooth newton method which is equivalent to an active set strategy a numerical example is given which confirms the theoretical results

serrated edge design and field analysis in spatial time and angular domains for improving the field uniformity of beam combiner edge diffraction of a beam combiner will adversely affect the field uniformity in the observation plane in order to suppress the edge diffraction the serrated edge design method is proposed according to the equivalence principle a field uniformity analysis method was proposed to compute the spatial field distribution radiated from the aperture of a beam combiner in order to separate the direct wave from diffracted wave the spatial field is transformed into the time and angular domain spectrum by inverse fourier transform first the height of serrations is designed by analog of the equivalent trapezoidal amplitude taper then the number of serrations was designed by the assumption of isosceles and right triangles finally the shape of serrations was designed by changing the interior angle of serrations a serrated edge design example is also proposed and analyzed by comparing the field uniformity in spatial time and angular domains before and after serrating the edges the amplitude and phase ripples decrease from 1 85 db and 12 16 to 0 53 db and 3 98 respectively the results show that the field uniformity is significantly improved

recovery of the transient heat transfer coefficient in the nonlinear boundary value problem for the heat equation the problem of numerical identification of the transient heat boundary coefficient for the one dimensional heat equation in a finite slab under nonlinear boundary conditions is examined on the basis of additional information regarding the solution after measuring the temperature and the heat flux at the non active boundary the inverse heat conduction problem is solved using a fully explicit and stable space marching scheme based on a finite difference implementation of the mollification method and the transient heat boundary coefficient is then approximately determined stability bounds are derived an several numerical examples to validate the technique provided

error estimates for variational normal derivatives and dirichlet control problems with energy regularization this article deals with error estimates for the finite element approximation of variational normal derivatives and as a consequence error estimates for the finite element approximation of dirichlet boundary control problems with energy regularization the regularity of the solution is carefully carved out exploiting weighted sobolev and holder spaces this allows to derive a sharp relation between the convergence rates for the approximation and the structure of the geometry more precisely the largest opening angle at the vertices of polygonal domains numerical experiments confirm that the derived convergence rates are sharp

error analysis for a finite element approximation of elliptic dirichlet boundary control problems we consider the galerkin finite element approximation of an elliptic dirichlet boundary control model problem governed by the laplacian operator the analytical setting of this problem uses l 2 controls and a very weak formulation of the state equation however the corresponding finite element approximation uses standard continuous trial and test functions for this approximation we derive a priori error estimates of optimal order which are confirmed by numerical experiments the proofs employ duality arguments and known results from the l p error analysis for the finite element dirichlet projection

on the regularity of the solutions of dirichlet optimal control problems in polygonal domains a linear quadratic dirichlet control problem governed by an elliptic equation posed on a possibly nonconvex polygonal domain is analyzed detailed regularity results are provided in classical sobolev slobodetski spaces in particular it is proved that in the presence of pointwise control constraints the optimal control is continuous despite the nonconvexity of the domain

boundary velocity control of incompressible flow with an application to viscous drag reduction an optimal boundary control problem for the navier stokes equations is presented the control is the velocity on the boundary which is constrained to lie in a closed convex subset of h 1 2 of the boundary a necessary condition for optimality is derived computations are done when the control set is actually finite dimensional resulting in an application to viscous drag reduction

an integral solution for the inverse heat conduction problem after the method of weber abstract we consider the linear inverse heat conduction problem ihcp in the one dimensional semi infinite slab and present a new solution algorithm and error bounds based on the integral representation of weber s hyperbolic approximation and a suitable filtered version of the noisy data we discuss the determination of the hyperbolic parameter and describe several numerical examples of interest showing the accuracy and stability of the method

existence of optimal controls for viscous flow problems a class of optimal control problems in viscous flow is studied main result is the existence theorem for optimal control three typical flow control problems are formulated within this general class

on existence analysis of steady flows of generalized newtonian fluids with concentration dependent power law index abstract we study a system of partial differential equations describing a steady flow of an incompressible generalized newtonian fluid wherein the cauchy stress is concentration dependent namely we consider a coupled system of the generalized navier stokes equations and convection diffusion equation with non linear diffusivity we prove the existence of a weak solution for certain class of models by using a generalization of the monotone operator theory which fits into the framework of generalized sobolev spaces with variable exponent such a framework is involved since the function spaces where we look for the weak solution are dependent of the solution itself and thus we a priori do not know them this leads us to the principal a priori assumptions on the model parameters that ensure the holder continuity of the variable exponent we present here a constructive proof based on the galerkin method that allows us to obtain the result for very general class of models

energy corrected fem for optimal dirichlet boundary control problems in the presence of re entrant corners the solution of pde constrained optimization problems in general has singular components even when the given data are smooth consequently the use of standard finite element discretization gives rise to the so called pollution effect this means that only a reduced convergence rate compared to the best approximation error is obtained we discuss how optimal convergence rates in weighted norms can be achieved using the idea of energy corrected finite element methods applied to optimal dirichlet boundary control problem in the energy space we present optimal error estimates in weighted norms for the state variable and for the control several numerical examples illustrate the obtained theoretical results

numerical approximation of optimal flow control problems by a penalty method error estimates and numerical results the purpose of this paper is to present numerically convenient approaches to solve optimal dirichlet control problems governed by the steady navier stokes equations we will examine a penalized neumann control approach for solving dirichlet control problems from numerical and computational points of view the control is affected by the suction or injection of fluid through the boundary or by boundary surface movements in the tangential direction the control objective is to minimize the vorticity in the flow or to drive the velocity field to a desired one we develop sequential quadratic programming methods to solve these optimal control problems the effectiveness of the optimal control techniques in flow controls and the feasibility of the proposed penalized neumann control approaches for flow control problems are demonstrated by numerical experiments for a viscous incompressible fluid flow in a two dimensional channel and in a cavity geometry

analysis and finite element approximation of optimal control problems for the stationary navier stokes equations with distributed and neumann controls we examine certain analytic and numerical aspects of optimal control problems for the stationary navier stokes equations the controls considered may be of either the distributed or neumann type the functionals minimized are either the viscous dissipation or the l4 distance of candidate flows to some desired flow we show the existence of optimal solutions and justify the use of lagrange multiplier techniques to derive a system of partial differential equations from which optimal solutions may be deduced we study the regularity of solutions of this system then we consider the approximation by finite element methods of solutions of the optimality system and derive optimal error estimates

an optimal control problem in exterior hydrodynamics in this paper we consider the problem of accelerating an obstacle in an incompressible viscous fluid from rest to a given speed in a given time with minimum energy expenditure an existence theorem for the speed trajectory which corresponds to the absolute minimum is provided the results are valid for arbitrary reynolds numbers

a superconvergent hybridizable discontinuous galerkin method for dirichlet boundary control of elliptic pdes we begin an investigation of hybridizable discontinuous galerkin hdg methods for approximating the solution of dirichlet boundary control problems governed by elliptic pdes these problems can involve atypical variational formulations and often have solutions with low regularity on polyhedral domains these issues can provide challenges for numerical methods and the associated numerical analysis we propose an hdg method for a dirichlet boundary control problem for the poisson equation and obtain optimal a priori error estimates for the control specifically under certain assumptions for a 2d convex polygonal domain we show the control converges at a superlinear rate we present 2d and 3d numerical experiments to demonstrate our theoretical results

numerical method for the solution of non linear two dimensional inverse heat conduction problem using unstructured meshes a new method of solving multidimensional heat conduction problems is formulated the developed space marching method allows to determine quickly and exactly unsteady temperature distributions in the construction elements of irregular geometry the method which is based on temperature measurements at the outer surface is especially appropriate for determining transient temperature distribution in thick wall pressure components two examples are included to demonstrate the capabilities of the new approach copyright 2000 john wiley sons ltd

multigrid algorithms for variational inequalities we present an iterative scheme for obstacle problems described by variational inequalities at each step of the iteration the method leads to a reduced linear resp nonlinear algebraic system whi

solution of inverse heat conduction problems with phase changes by the mollification method abstract the mollification method originally developed for the solution of the inverse heat conduction problem is implemented as a front tracking space marching finite difference procedure to determine the location of the solid liquid interface in one dimensional melting problems the technique is reversible in the sense that if the temperature history is approximately known at two arbitrary points in the liquid region or in the solid region the stefan problem is solved by marching forward or backward in space the method is also applied to the solution of the inverse stefan problem where the temperature and heat flux histories are measured at the transient front interface and the temperature and heat flux histories at the fixed boundary are desired the accuracy of the algorithm is illustrated by several linear and nonlinear examples where exact and noisy data are considered

necessary and sufficient conditions for optimal controls in viscous flow problems a class of optimal control problems in viscous flow is studied main results are the pontryagin maximum principle and the verification theorem for the hamilton jacobi bellman equation characterising the feedback problem the maximum principle is established by two quite different methods

parameter selection by discrete mollification and the numerical solution of the inverse heat conduction problems abstract a procedure for the numerical solution of the one dimensional inverse heat conduction problem based on the computaion of the solution associated with a suitable filtered version of the noisy data by discrete mollification is presented and a parameter choice criterion which automatically determines the radius of mollification as a function of the amount of noise in the data is introduced several numerical examples of interest are also analyzed showing the accuracy and stability properties of the method

a reduced order method for simulation and control of fluid flows this article presents a reduced order modeling approach for simulation and control of viscous incompressible flows the reduced order models suitable for control and which capture the essential physics are developed using the reduced basis method the so called lagrange approach is used to define reduced bases and the basis functions in this approach are obtained from the numerical solutions the feasibility of this method for flow control is demonstrated on boundary control problems in closed cavity and in wall bounded channel flows control action is effected through boundary surface movement on a part of the solid wall our formulation of the reduced order method applied to flow control problems leads to a constrained minimization problem and is solved by applying newton like methods to the necessary conditions of optimality through our computational experiments we demonstrate the feasibility and applicability of the reduced order method for simulation and control of fluid flows

wavelet and fourier methods for solving the sideways heat equation we consider an inverse heat conduction problem the sideways heat equation which is a model of a problem where one wants to determine the temperature on both sides of a thick wall but where one side is inaccessible to measurements mathematically it is formulated as a cauchy problem for the heat equation in a quarter plane with data given along the line x 1 where the solution is wanted for 0 leq x 1 the problem is ill posed in the sense that the solution if it exists does not depend continuously on the data we consider stabilizations based on replacing the time derivative in the heat equation by wavelet based approximations or a fourier based approximation the resulting problem is an initial value problem for an ordinary differential equation which can be solved by standard numerical methods e g a runge kutta method we discuss the numerical implementation of fourier and wavelet methods for solving the sideways heat equation theory predicts that the fourier method and a method based on meyer wavelets will give equally good results our numerical experiments indicate that also a method based on daubechies wavelets gives comparable accuracy as test problems we take model equations with constant and variable coefficients we also solve a problem from an industrial application with actual measured data

semidiscrete approximations of optimal robin boundary control problems constrained by semilinear parabolic pde an optimal robin boundary control problem associated with semilinear parabolic partial differential equations is considered existence of an optimal solution is proved and an optimality system of equations is derived semidiscrete finite element approximations of the optimality system are defined and error estimates are obtained

error estimates for the numerical approximation of dirichlet boundary control for semilinear elliptic equations we study the numerical approximation of boundary optimal control problems governed by semilinear elliptic partial differential equations with pointwise constraints on the control the control is the trace of the state on the boundary of the domain which is assumed to be a convex polygonal open set in mathbb r 2 piecewise linear finite elements are used to approximate the control as well as the state we prove that the error estimates are of order o h 1 1 p for some p 2 which is consistent with the w 1 1 p p gamma regularity of the optimal control

numerical identification of interface source functions in heat transfer problems under nonlinear boundary conditions abstract an inverse approach based on a fully explicit and unconditionally stable finite difference implementation of the mollification method for the approximate identification of transient interface sources in semi infinite or finite slab mediums subjects to nonlinear boundary conditions is presented stability bounds are derived an two typical examples corresponding to newton cooling law and stefan boltzmann radiation law are illustrated

dynamic programming of the navier stokes equations a class of optimal control problems in viscous flow is studied main results are the pontryagin maximum principle and the verification theorem for the hamilton jacobi bellman equation

optimization problems for the navier stokes equations with regular boundary controls abstract we study the necessary conditions for an optimal boundary control problem associated with the stationary navier stokes equations with regular controls the control is the velocity on part or all of the boundary of the given flow domain we present a rigorous justification for the use of the lagrange multiplier rule to derive first order necessary conditions for optimality these are expressed as a system of partial differential equations we study the regularity of solutions of this system and finally we give some examples of specific functionals useful in applications

dynamics for controlled navier stokes systems with distributed controls the long time behavior of solutions for an optimal distributed control problem associated with the navier stokes equations is studied first a linear feedback solution for the navier stokes equations is constructed this feedback solution possesses decay in time properties then some preliminary estimates for the long time behavior of all solutions of the navier stokes equations are derived next the existence of a solution for the optimal control problem is proved finally the long time decay properties for the optimal solutions are established

a c0 interior penalty method for the dirichlet control problem governed by biharmonic operator an energy space based dirichlet boundary control problem governed by biharmonic equation is investigated and subsequently a c0 interior penalty method is proposed and analyzed an abstract a priori error estimate is derived under the minimal regularity conditions the abstract error estimate guarantees optimal order of convergence whenever the solution is sufficiently regular further an optimal order l2 norm error estimate is derived numerical experiments illustrate the theoretical findings

boundary value problems and optimal boundary control for the navier stokes system the two dimensional case we study optimal boundary control problems for the two dimensional navier stokes equations in an unbounded domain control is effected through the dirichlet boundary condition and is sought in a subset of the trace space of velocity fields with minimal regularity satisfying the energy estimates an objective of interest is the drag functional we first establish three important results for inhomogeneous boundary value problems for the navier stokes equations namely we identify the trace space for the velocity fields possessing finite energy we prove the existence of a solution for the navier stokes equations with boundary data belonging to the trace space and we identify the space in which the stress vector along the boundary of admissible solutions is well defined then we prove the existence of an optimal solution over the control set finally we justify the use of lagrange multiplier principles derive an optimality system of equations in the weak sense from which optimal states and controls may be determined and prove that the optimality system of equations satisfies in appropriate senses a system of partial differential equations with boundary values

compact range reflector analysis using the plane wave spectrum approach with an adjustable sampling rate an improved method for determining the test zone field of compact range reflectors is presented the plane wave spectrum pws approach is used to obtain the test zone field from knowledge of the reflector aperture field distribution the method is particularly well suited to the analysis of reflectors with a linearly serrated rim for reduced edge diffraction computation of the pws of the reflector aperture field is facilitated by a closed form expression for the fourier transform of a polygonal window function inverse transformation in the test zone region is accomplished using a fast fourier transform fft algorithm with a properly adjusted sampling rate which is a function of both the reflector size and the distance from the reflector the method is validated by comparison with results obtained using surface current and aperture field integration techniques the performance of several serrated reflectors is evaluated in order to observe the effects of edge diffraction on the test zone fields

a stable space marching finite differences algorithm for the inverse heat conduction problem with no initial filtering procedure abstract we consider the one dimensional semi infinite linear inverse heat conduction problem ihcp and present a new solution algorithm based on singular perturbation techniques the new method successfully incorporates the needed initial filtering procedure of the noisy data into the marching scheme the concept of formal stability for the ihcp is analyzed and its relationship with the high frequency components of the measured data funciton is further studied by comparing the new method with the mollification method and the method of weber a fully explicit space marching finite differences scheme is developed and several numerical experiments of interest showing the accuracy and stability properties of the algorithm in the presence of noisy data and for small dimensionless time steps are investigated

finite element approximation of dirichlet boundary control for elliptic pdes on two and three dimensional curved domains we consider the variational discretization of elliptic dirichlet optimal control problems with constraints on the control the underlying state equation which is considered on smooth two and three dimensional domains is discretized by linear finite elements taking into account domain approximation the control variable is not discretized we obtain optimal error bounds for the optimal control in two and three space dimensions and prove a superconvergence result in two dimensions provided that the underlying mesh satisfies some additional condition we confirm our analytical findings by numerical experiments

superconvergence properties of optimal control problems an optimal control problem for a two dimensional 2 d elliptic equation is investigated with pointwise control constraints this paper is concerned with discretization of the control by piecewise constant functions the state and the adjoint state are discretized by linear finite elements approximations of the optimal solution of the continuous optimal control problem will be constructed by a projection of the discrete adjoint state it is proved that these approximations have convergence order h2

a stable approach to solving one dimensional inverse problems the study of one dimensional inverse scattering problems leads to certain fredholm integral equations of the first kind as is usually the case unless great care is taken in the discretization process an ill conditioned algebraic system results by taking advantage of special characteristics of the kernel it is shown that a judicious choice of mesh leads to very well conditioned systems moreover the solution of the discrete system is shown to be effectively the expansion of the actual solution in a generalized fourier series

a numerical method for the solution of two dimensional inverse heat conduction problems a numerical method for the solution of inverse heat conduction problems in two dimensional rectangular domains is established and its performance is demonstrated by computational results the present method extends beck s8 method to two spatial dimensions and also utilizes future times in order to stabilize the ill posedness of the underlying problems the approach relies on a line approximation of the elliptic part of the parabolic differential equation leading to a system of one dimensional problems which can be decoupled