on the stabilization of a flexible beam with a tip mass we study the stability of a flexible beam that is clamped at one end and free at the other a mass is also attached to the free end of the beam to stabilize this system we apply a boundary control force at the free end of the beam we prove that the closed loop system is well posed and is exponentially stable we then analyze the spectrum of the system for a special case and prove that the spectrum determines the exponential decay rate for the considered case

a fast poisson solver for complex geometries robust fast solvers for the poisson equation have generally been limited to regular geometries where direct methods based on fourier analysis or cyclic reduction and multigrid methods can be used while multigrid methods can be applied in irregular domains and to a broader class of partial differential equations they are difficult to implement in a robust fashion since they require an appropriate hierarchy of coarse grids which are not provided in many practical situations in this paper we present a new fast poisson solver based on potential theory rather than on direct discretization of the partial differential equation our method combines fast algorithms for computing volume integrals and evaluating layer potentials on a grid with a fast multipole accelerated integral equation solver the amount of work required is o m log m n where m is the number of interior grid points and n is the number of points on the boundary asymptotically the cost of our method is just twice that of a standard poisson solver on a rectangular domain in which the problem domain can be embedded independent of the complexity of the geometry

asymptotic behavior of interior point methods a view from semi infinite programming we study the asymptotic behavior of interior point methods for linear programming problems attempts to solve larger problems using interior point methods lead to the question of how these algorithms behave as n the number of variables goes to infinity here we take a different point of view and investigate what happens when n is infinite motivated by this approach we study the limits of search directions potential functions and central paths we also suggest that the complexity of some linear programming problems may depend on the smoothness of the given problem rather than the number of variables we prove that when n is infinite for some subclasses of problems one can still obtain a bound on the number of iterations required in terms of the smoothness of the problem and the desired accuracy

convergence behavior of interior point algorithms we show that most interior point algorithms for linear programming generate a solution sequence in which every limit point satisfies the strict complementarity condition these algorithms include all path following algorithms and some potential reduction algorithms the result also holds for the monotone complementarity problem if a strict complementarity solution exists in general the limit point is a solution that maximizes the number of its nonzero components among all solutions

exponential stability for a class of boundary conditions on a euler bernoulli beam subject to disturbances via boundary control abstract this paper outlines the procedure for applying a sliding mode backstepping boundary control technique to a euler bernoulli beam subject to unknown bounded disturbances on the boundary for four different types of boundary conditions referred to in this paper as a class of boundary conditions these boundary conditions include all combinations of pinned and sliding types with control and disturbances on one boundary i e pinned pinned pinned sliding sliding pinned and sliding sliding with input on the second boundary the technique was developed for the specific case of a pinned pinned beam in the literature and in this paper the technique is generalized to the degree that it is possible to cover all four of the aforementioned cases furthermore a proof of the fully arbitrary exponential stability of the closed loop system is provided in this paper to this point only asymptotic stability was shown for the closed loop system for one case the technique outlined in this paper provides for a class of boundary conditions a controller that exponentially stabilizes the vibrations in a beam in the presence of bounded unknown disturbances the beam displacement returns to the origin and its decay is bounded by a known exponential decay function after a known finite time reaching phase

a second order accurate symmetric discretization of the poisson equation on irregular domains in this paper we consider the variable coefficient poisson equation with dirichlet boundary conditions on an irregular domain and show that one can obtain second order accuracy with a rather simple discretization moreover since our discretization matrix is symmetric it can be inverted rather quickly as opposed to the more complicated nonsymmetric discretization matrices found in other second order accurate discretizations of this problem multidimensional computational results are presented to demonstrate the second order accuracy of this numerical method in addition we use our approach to formulate a second order accurate symmetric implicit time discretization of the heat equation on irregular domains then we briefly consider stefan problems

a review of algebraic multigrid abstract since the early 1990s there has been a strongly increasing demand for more efficient methods to solve large sparse unstructured linear systems of equations for practically relevant problem sizes classical one level methods had already reached their limits and new hierarchical algorithms had to be developed in order to allow an efficient solution of even larger problems this paper gives a review of the first hierarchical and purely matrix based approach algebraic multigrid amg amg can directly be applied for instance to efficiently solve various types of elliptic partial differential equations discretized on unstructured meshes both in 2d and 3d since amg does not make use of any geometric information it is a plug in solver which can even be applied to problems without any geometric background provided that the underlying matrix has certain properties

an example on the effect of time delays in boundary feedback stabilization of wave equations this note is concerned with the erect of time delays in boundary feedback stabilization schemes for wave equations the question to be addressed is whether such delays can destabilize a system which is uniformly asymptotically stable in the absence of delays

an efficient algorithm for hydrodynamical interaction of many deformable drops an efficient and accurate 3d algorithm for dynamical simulations of many deformable drops with strong hydrodynamical interactions at zero reynolds numbers is developed the drop to medium viscosity ratio and the bond number are arbitrary and the drops are subject to gravity with stationary triply periodic boundary conditions the algorithm at each step is a hybrid of boundary integral and economical multipole techniques with extensive use of rotational transformations and economical truncation of multipole expansions to optimize near field interactions a significant part of the code is the new best paraboloid spline technique for calculating the normal vectors and curvatures on drop surfaces which greatly improves the quality of long time simulations examples show the phenomenon of clustering in a concentrated sedimenting emulsion for 0 25 and 1 which leads to an increase in the average sedimentation velocity with time a high efficiency of the method is demonstrated with two orders of magnitude gains over the standard o n2n2 boundary integral technique for n 102 drops in a periodic cell with n 103 triangular boundary elements per drop so that typical long time dynamical simulations can be performed in a few days or weeks on a standard workstation as compared to the several years which would be required for the same simulations using standard boundary integral techniques the effects of drop triangulation and truncation of multipole expansions on dynamical simulations are assessed

a fluctuating boundary integral method for brownian suspensions abstract we present a fluctuating boundary integral method fbim for overdamped brownian dynamics bd of two dimensional periodic suspensions of rigid particles of complex shape immersed in a stokes fluid we develop a novel approach for generating brownian displacements that arise in response to the thermal fluctuations in the fluid our approach relies on a first kind boundary integral formulation of a mobility problem in which a random surface velocity is prescribed on the particle surface with zero mean and covariance proportional to the green s function for stokes flow stokeslet this approach yields an algorithm that scales linearly in the number of particles for both deterministic and stochastic dynamics handles particles of complex shape achieves high order of accuracy and can be generalized to three dimensions and other boundary conditions we show that brownian displacements generated by our method obey the discrete fluctuation dissipation balance relation dfdb based on a recently developed positively split ewald method fiore et al 2017 24 near field contributions to the brownian displacements are efficiently approximated by iterative methods in real space while far field contributions are rapidly generated by fast fourier space methods based on fluctuating hydrodynamics fbim provides the key ingredient for time integration of the overdamped langevin equations for brownian suspensions of rigid particles we demonstrate that fbim obeys dfdb by performing equilibrium bd simulations of suspensions of starfish shaped bodies using a random finite difference temporal integrator

finding the projection on a polytope an iterative method abstract an iterative algorithm is developed for the problem of finding the projection of a point on a polytope described as intersection of half spaces a proof of the convergence is given and its efficiency is investigated

constant potential primal dual algorithms a framework we start with a study of the primal dual affine scaling algorithms for linear programs using ideas from kojima et al mizuno and nagasawa and new potential functions we establish a framework for primal dual algorithms that keep a potential function value fixed we show that if the potential function used in the algorithm is compatible with a corresponding neighborhood of the central path then the convergence proofs simplify greatly our algorithms have the property that all the iterates can be kept in a neighborhood of the central path without using any centering in the search directions

an o n l iteration bound primal dual cone affine scaling algorithm for linear programming in this paper we introduce a primal dual affine scaling method the method uses a search direction obtained by minimizing the duality gap over a linearly transformed conic section this direction neither coincides with known primal dual affine scaling directions jansen et al 1993 monteiro et al 1990 nor does it fit in the generic primal dual method kojima et al 1989 the new method requires o sqrt n l main iterations it is shown that the iterates follow the primal dual central path in a neighbourhood larger than the conventional mathcal n 2 neighbourhood the proximity to the primal dual central path is measured by trigonometric functions

immersed interface methods for stokes flow with elastic boundaries or surface tension a second order accurate interface tracking method for the solution of incompressible stokes flow problems with moving interfaces on a uniform cartesian grid is presented the interface may consist of an elastic boundary immersed in the fluid or an interface between two different fluids the interface is represented by a cubic spline along which the singularly supported elastic or surface tension force can be computed the stokes equations are then discretized using the second order accurate finite difference methods for elliptic equations with singular sources developed in our previous paper siam j numer anal 31 1994 pp 1019 1044 the resulting velocities are interpolated to the interface to determine the motion of the interface an implicit quasi newton method is developed that allows reasonable time steps to be used

a three dimensional computational method for blood flow in the heart 1 immersed elastic fibers in a viscous incompressible fluid abstract this paper describes the numerical solution of the 3 dimensional equations of motion of a viscous incompressible fluid that contains an immersed system of elastic fibers implementation details such as vectorization and the efficient use of external memory are discussed the method is applied to the damped vibrations of a fiber wound toroidal tube and empirical evidence of convergence is presented

superlinear primal dual affine scaling algorithms for lcp we describe an interior point algorithm for monotone linear complementarity problems in which primal dual affine scaling is used to generate the search directions the algorithm is shown to have global and superlinear convergence with q order up to but not including two the technique is shown to be consistent with a potential reduction algorithm yielding the first potential reduction algorithm that is both globally and superlinearly convergent

rapid solution of integral equations of classical potential theory an algorithm is described for rapid solution of classical boundary value problems dirichlet an neumann for the laplace equation based on iteratively solving integral equations of potential theory cpu time requirements for previously published algorithms of this type are proportional to n2 where n is the number of nodes in the discretization of the boundary of the region the cpu time requirements for the algorithm of the present paper are proportional to n making it considerably more practical for large scale problems

integral equation methods for stokes flow and isotropic elasticity in the plane we present a class of integral equation methods for the solution of biharmonic boundary value problems with applications to two dimensional stokes flow and isotropic elasticity the domains may be multiply connected and finite infinite or semi infinite in extent our analytic formulation is based on complex variables and our fast multipole based iterative solution procedure requireso n operations wherenis the number of nodes in the discretization of the boundary the performance of the methods is illustrated with several large scale numerical examples

evaluation of single layer potentials over curved surfaces the evaluation of nearly singular single layer potentials encountered in boundary element methods is treated by a new approach the potential is expressed as a sum of a one dimensional integral and a correction term that vanishes for planar surfaces the small variance of the second term s integrand allows the use of a quasi monte carlo quadrature numerical results show that a significant reduction in computational time is obtained over algorithms employing domain subdivisions

a cartesian grid embedded boundary method for poisson s equation on irregular domains we present a numerical method for solving poisson s equation with variable coefficients and dirichlet boundary conditions on two dimensional regions the approach uses a finite volume discretization which embeds the domain in a regular cartesian grid we treat the solution as a cell centered quantity even when those centers are outside the domain cells that contain a portion of the domain boundary use conservative differencing of second order accurate fluxes on each cell volume the calculation of the boundary flux ensures that the conditioning of the matrix is relatively unaffected by small cell volumes this allows us to use multigrid iterations with a simple point relaxation strategy we have combined this with an adaptive mesh refinement amr procedure we provide evidence that the algorithm is second order accurate on various exact solutions and compare the adaptive and nonadaptive calculations

immersed interface methods for neumann and related problems in two and three dimensions we develop and apply a finite difference method to discretize the laplacian operator with neumann boundary conditions on an irregular domain using a regular cartesian grid the method is an extension of the immersed interface method developed by leveque and li siam j numer anal 31 1994 1019 1044 with careful selection of stencils the method is second order accurate and produces a matrix that is stable diagonally semidominant the method is illustrated on several two dimensional problems and one three dimensional problem

high order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources this paper introduces a novel high order interface scheme the matched interface and boundary mib method for solving elliptic equations with discontinuous coefficients and singular sources on cartesian grids by appropriate use of auxiliary line and or fictitious points physical jump conditions are enforced at the interface unlike other existing interface schemes the proposed method disassociates the enforcement of physical jump conditions from the discretization of the differential equation under study to construct higher order interface schemes the proposed mib method bypasses the major challenge of implementing high order jump conditions by repeatedly enforcing the lowest order jump conditions the proposed mib method is of arbitrarily high order in principle in treating straight regular interfaces we construct mib schemes up to 16th order for more general elliptic problems with curved irregular interfaces and boundary up to 6th order mib schemes have been demonstrated by employing the standard high order finite difference schemes to discretize the laplacian the present mib method automatically reduces to the standard central difference scheme when the interface is absent the immersed interface method iim is regenerated for a comparison study of the proposed method the robustness of the mib method is verified against the large magnitude of the jump discontinuity across the interface the nature of high efficiency and low memory requirement of the mib method is extensively validated via solving various elliptic immersed interface problems in two and three dimensions ee dimensions

the immersed interface method for elliptic equations with discontinuous coefficients and singular sources the authors develop finite difference methods for elliptic equations of the form nabla cdot beta x nabla u x kappa x u x f x in a region omega in one or two space dimension

the fast solution of poisson s and the biharmonic equations on irregular regions we present fast methods for solving laplace s and the biharmonic equations on irregular regions with smooth boundaries the methods used for solving both equations make use of fast poisson solvers on a rectangular region in which the irregular region is embedded they also both use an integral equation formulation of the problem where the integral equations are fredholm integral equations of the second kind the main idea is to use the integral equation formulation to define a discontinuous extension of the solution to the rest of the rectangular region fast solvers are then used to compute the extended solution aside from solving the equations we have also been able to compute derivatives of the solutions with little loss of accuracy when the data was sufficiently smooth

a fast solver for the stokes equations with distributed forces in complex geometries we present a new method for the solution of the stokes equations the main features of our method are 1 it can be applied to arbitrary geometries in a black box fashion 2 it is second order accurate and 3 it has optimal algorithmic complexity our approach to which we refer as the embedded boundary integral method ebi is based on anita mayo s work for the poisson s equation the fast solution of poisson s and the biharmonic equations on irregular regions siam journal on numerical analysis 21 1984 285 299 we embed the domain in a rectangular domain for which fast solvers are available and we impose the boundary conditions as interface jump conditions on the velocities and tractions we use an indirect boundary integral formulation for the homogeneous stokes equations to compute the jumps the resulting equations are discretized by nystrom s method the rectangular domain problem is discretized by finite elements for a velocity pressure formulation with equal order interpolation bilinear elements q1 q1 stabilization is used to circumvent the inf sup condition for the pressure space for the integral equations fast matrix vector multiplications are achieved via an n logn algorithm based on a block representation of the discrete integral operator combined with kernel independent singular value decomposition to sparsify low rank blocks the regular grid solver is a krylov method conjugate residuals combined with an optimal two level schwartz preconditioner for the integral equation we use gmres we have tested our algorithm on several numerical examples and we have observed optimal convergence rates

potential reduction methods in mathematical programming we provide a survey of interior point methods for linear programming and its extensions that are based on reducing a suitable potential function at each iteration we give a fairly complete overview of potential reduction methods for linear programming focusing on the possibility of taking long steps and the properties of the barrier function that are necessary for the analysis we then describe briefly how the methods and results can be extended to certain convex programming problems following the approach of nesterov and todd we conclude with some open problems

a fast iterative algorithm for elliptic interface problems a fast second order accurate iterative method is proposed for the elliptic equation grad cdot beta x y grad u f x y in a rectangular region omega in two space dimensions we assume that there is an irregular interface across which the coefficient beta the solution u and its derivatives and or the source term f may have jumps we are especially interested in the cases where the coefficients beta are piecewise constant and the jump in beta is large the interface may or may not align with an underlying cartesian grid the idea in our approach is to precondition the differential equation before applying the immersed interface method proposed by leveque and li siam j numer anal 4 1994 pp 1019 1044 in order to take advantage of fast poisson solvers on a rectangular region an intermediate unknown function the jump in the normal derivative across the interface is introduced our discretization is equivalent to using a second order difference scheme for a corresponding poisson equation in the region and a second order discretization for a neumann like interface condition thus second order accuracy is guaranteed a gmres iteration is employed to solve the schur complement system derived from the discretization a new weighted least squares method is also proposed to approximate interface quantities from a grid function numerical experiments are provided and analyzed the number of iterations in solving the schur complement system appears to be independent of both the jump in the coefficient and the mesh size

active boundary control of an euler bernoulli beam for generating vibration free state abstract this paper presents active boundary control abc of an euler bernoulli beam which enables one to generate a desired boundary condition at any designated position of a target beam structure thereby permitting the structure to possess desired properties characterized by the boundary condition furthermore abc has potential to create a completely vibration free state in the designated area of a beam this paper begins by presenting the principle of abc using a transfer matrix method the optimal control law of the abc system being derived it is found that in addition to conventional four classical boundary conditions free pinned clamped and sliding support abc can generate two more boundary conditions that may not be observed in real systems but realized by abc it is also found that as a result of applying abc to a specific location including a current conventional boundary of a beam a completely vibration free state in the target region of a beam can be realized finally an experiment using an adaptive feedforward control was conducted demonstrating that abc enables the generation of a desired boundary condition at the designated location of a target beam and of a completely vibration free state of a beam

a polynomial time primal dual affine scaling algorithm for linear and convex quadratic programming and its power series extension we describe an algorithm for linear and convex quadratic programming problems that uses power series approximation of the weighted barrier path that passes through the current iterate in order to find the next iterate if r 1 is the order of approximation used we show that our algorithm has time complexity o n 1 1 r l 1 1 r iterations and o n3 n2r arithmetic operations per iteration where n is the dimension of the problem and l is the size of the input data when r 1 we show that the algorithm can be interpreted as an affine scaling algorithm in the primal dual setup

rapid parallel evaluation of integrals in potential theory on general three dimensional regions we present a new high order accurate method for the rapid parallel evaluation of certain integrals in potential theory on general three dimensional regions these methods use fast methods for solving the differential equation which the kernel satisfies and the number of operations needed to evaluate the integrals is essentially equal to the number of operations needed to solve the differential equation on a regular rectangular grid in particular one can evaluate integrals whose kernels are the greens function for poissons equation by using fourier methods on a rectangular grid or a fast poisson solver thus these methods avoid the problems associated with using quadrature methods to evaluate an integral with a singular kernel numerical results are presented for experiments on a variety of geometries

a fictitious domain method for external incompressible viscous flow modeled by navier stokes equations abstract in this article we discuss a fictitious domain method for the numerical solutions of three dimensional elliptic problems with dirichlet boundary conditions and also of the navier stokes equations modeling incompressible viscous flow the methodology for the navier stokes equations described here takes a systematic advantage of time discretization by operator splitting in order to treat separately advection imbedding and incompressibility due to the decoupling fast elliptic solvers can be used to treat the incompressibility condition even if the original problem is taking place on a nonregular geometry the resulting methodology is applied to two dimensional unsteady external incompressible viscous flow problems and three dimensional stokes problems

the immersed interface method for the navier stokes equations with singular forces abstract peskin s immersed boundary method has been widely used for simulating many fluid mechanics and biology problems one of the essential components of the method is the usage of certain discrete delta functions to deal with singular forces along one or several interfaces in the fluid domain however the immersed boundary method is known to be first order accurate and usually smears out the solutions in this paper we propose an immersed interface method for the incompressible navier stokes equations with singular forces along one or several interfaces in the solution domain the new method is based on a second order projection method with modifications only at grid points near or on the interface from the derivation of the new method we expect fully second order accuracy for the velocity and nearly second order accuracy for the pressure in the maximum norm including those grid points near or on the interface this has been confirmed in our numerical experiments furthermore the computed solutions are sharp across the interface nontrivial numerical results are provided and compared with the immersed boundary method meanwhile a new version of the immersed boundary method using the level set representation of the interface is also proposed in this paper

primal dual affine scaling interior point methods for linear complementarity problems a first order affine scaling method and two m th order affine scaling methods for solving monotone linear complementarity problems lcps are presented all three methods produce iterates in a wide neighborhood of the central path the first order method has o nl 2 lognl 2 log lognl 2 iteration complexity if the lcp admits a strict complementary solution then both the duality gap and the iteration sequence converge superlinearly with q order two if m omega log sqrt n l then both higher order methods have o sqrt n l iteration complexity the q order of convergence of one of the methods is m 1 for problems that admit a strict complementarity solution while the q order of convergence of the other method is m 1 2 for general monotone lcps

locally corrected multidimensional quadrature rules for singular functions accurate numerical integration of singular functions usually requires either adaptivity or product integration both interfere with fast summation techniques and thus hamper large scale computations this paper presents a method for computing highly accurate quadrature formulas for singular functions which combine well with fast summation methods given the singularity and the n nodes we first construct weights which integrate smooth functions with order k accuracy then we locally correct a small number of weights near the singularity to achieve order k accuracy on singular functions as well the method is highly efficient and runs in o nk 2d n log 2 n time and o k 2d n space we derive precise error bounds and time estimates and confirm them with numerical results which demonstrate the accuracy and efficiency of the method in large scale computations as part of our implementation we also construct a new adaptive multidimensional product gauss quadrature routine with an effective error

locally corrected semi lagrangian methods for stokes flow with moving elastic interfaces we present a new method for computing two dimensional stokes flow with moving interfaces that respond elastically to stretching the interface is moved by semi lagrangian contouring a distance function is introduced on a tree of cells near the interface transported by a semi lagrangian time step and then used to contour the new interface the velocity field in a periodic box is calculated as a potential integral resulting from interfacial and body forces using a technique based on ewald summation with analytically derived local corrections the interfacial stretching is found from a surprisingly natural formula a test problem with an exact solution is constructed and used to verify the speed accuracy and robustness of the approach

boundary control of three dimensional inextensible marine risers this paper present a design of boundary controllers actuated by hydraulic actuators at the top end for global stabilization of a three dimensional riser system first a set of partial and ordinary differential equations describing motion of the riser and hydraulic systems is developed second several important properties of the riser system are derived based on these properties we show that the conventional formula to calculate the riser effective tension is oversimplified and a new formula is provided next boundary controllers are designed based on lyapunov s direct method the backstepping technique the derived properties of the riser system dynamics and poincare s inequalities finally the galerkin approximation method is used to prove existence and uniqueness of the solutions of the closed loop control system

a fourth order kernel free boundary integral method for implicitly defined surfaces in three space dimensions abstract the kernel free boundary integral kfbi method is a finite difference version of the traditional boundary integral method for elliptic and parabolic partial differential equations on complex domains it evaluates boundary or volume integrals involved in the solution of boundary integral equations bies by solving equivalent but simple interface problems on regular grids so that the integral kernel or green function is never needed or computed this is the essential difference of the kfbi method from the traditional ones it takes advantage of the well conditioning property of discrete bies so that the number of krylov subspace iterations is essentially independent of discretization parameter or system dimension this paper presents a fourth order kernel free boundary integral method for second order elliptic partial differential equations on complex domains in three space dimensions whose boundaries are given by implicitly defined surfaces it represents the domain boundary and discretizes data on it by intersection points of the surface with cartesian grid lines the approach has a variety of advantages the current work solves simple interface problems with corrected 27 point compact finite difference schemes and calculates the discrete equations with a fast fourier transform based elliptic solver numerical examples show that the proposed method is efficient as well as accurate

active boundary control of elastic cables theory and experiment cables are lightweight structural elements used in a variety of engineering applications in this paper an active boundary control system is introduced that damps undesirable vibrations in a cable using hamilton s principle the governing non linear partial differential equations for an elastic cable are derived including the natural boundary conditions associated with boundary force control based on lyapunov theory passive and active vibration controllers are developed a galerkin approach generates the linearized closed loop modal dynamics equations for out of plane vibration simulations and experiments demonstrate the improved damping provided by the active boundary controller

a self adjusting interior point algorithm for linear complementarity problems abstract based on the min max principle the standard centering equation in the interior point method is replaced by the optimality condition of a new proximity measure function thus a self adjusting mechanism is constructed in the new perturbed system the newton direction can be adjusted self adaptively according to the information of last iterates a self adjusting interior point method is given based on the new perturbed system numerical comparison is made between this algorithm and a primal dual interior point algorithm using standard perturbed system results demonstrate the efficiency and some advantages of the proposed algorithm

improved volume conservation in the computation of flows with immersed elastic boundaries this paper introduces a recipe for the construction of a finite difference divergence operator the coefficients of which are completely determined once an interpolation scheme has been chosen substitution of this divergence operator for the previously used divergence based on central differences makes a dramatic improvement in the overall volume conservation that is observed in an immersed boundary computation this improvement is particularly important for computations in which an elastic boundary separates chambers containing fluid at substantially different pressures a situation which is prominent in cardiac fluid dynamics during the contraction of the ventricles

the explicit jump immersed interface method finite difference methods for pdes with piecewise smooth solutions many boundary value problems bvps or initial bvps have nonsmooth solutions with jumps along lower dimensional interfaces the explicit jump immersed interface method ejiim was developed following li s fast iterative immersed interface method fiiim recognizing that the foundation for the efficient solution of many such problems is a good solver for elliptic bvps ejiim generalizes the class of problems for which fiiim is applicable it handles interfaces between constant and variable coefficients and extends the immersed interface method iim to bvps on irregular domains with neumann and dirichlet boundary conditions proofs of second order convergence for a one dimensional 1d problem with piecewise constant coefficients and for two dimensional 2d problems with singular sources are given other problems are reduced to the singular sources case with additional equations determining the source strengths the advantages of ejiim are high quality of solutions even on coarse grids and easy adaptation to many problems with complicated geometries while still maintaining the efficiency of the fiiim

optimal control of power flow at structural junctions abstract this paper describes several techniques for deriving optimal feedback compensators for structural waveguides at junctions a frequency dependent cost functional composed of power flow and control effort is minimized control of power flow by modifying junction reflection and transmission properties enables incoming vibrational power to be selectively absorbed matched termination non causal causal fixed form and wiener hopf feedback solutions are derived these solutions including a positive real approximation to the wiener hopf solution are illustrated through an extensive example for the free end of a dispersive bernoulli euler beam several interesting results arise from this research a matched termination absorbing all impinging energy is a subset of the optimal non causal solution as a result performance ranging from that achieved with simple rate feedback to that exceeding the matched termination is possible with increasingly complex configurations of control hardware in the formulation of the control information about the spectral content of the incoming waves can be used to frequency tailor the control performance the wave mode control formulation reveals that performance can be improved by using more than one distinct actuator and sensor at the active junction a process not generally practiced in structural control finally given the appropriate control hardware and particular structural geometries optimal wave control can be used to eliminate resonant behavior or dynamically isolate one structural region from another

three dimensional matched interface and boundary mib method for treating geometric singularities this paper reports the three dimensional 3d generalization of our previous 2d higher order matched interface and boundary mib method for solving elliptic equations with discontinuous coefficients and non smooth interfaces new mib algorithms that make use of two sets of interface jump conditions are proposed to remove the critical acute angle constraint of our earlier mib scheme for treating interfaces with sharp geometric singularities such as sharp edges sharp wedges and sharp tips the resulting 3d mib schemes are of second order accuracy for arbitrarily complex interfaces with sharp geometric singularities of fourth order accuracy for complex interfaces with moderate geometric singularities and of sixth order accuracy for curved smooth interfaces a systematical procedure is introduced to make the mib matrix optimally symmetric and banded by appropriately choosing auxiliary grid points consequently the new mib linear algebraic equations can be solved with fewer number of iterations the proposed mib method makes use of cartesian grids standard finite difference schemes lowest order interface jump conditions and fictitious values the interface jump conditions are enforced at each intersecting point of the interface and mesh lines to overcome the staircase phenomena in finite difference approximation while a pair of fictitious values are determined along a mesh at a time an iterative procedure is proposed to determine all the required fictitious values for higher order schemes by repeatedly using the lowest order jump conditions a variety of mib techniques are developed to overcome geometric constraints the essential strategy of the mib method is to locally reduce a 2d or a 3d interface problem into 1d like ones the proposed mib method is extensively validated in terms of the order of accuracy the speed of convergence the number of iterations and cpu time numerical experiments are carried out to complex interfaces including the molecular surfaces of a protein a missile interface and van der waals surfaces of intersecting spheres

boundary control of transverse motion of marine risers with actuator dynamics this paper presents a boundary controller to reduce transverse motion of flexible marine risers driven by a hydraulic system at the top end of the risers under environmental disturbances induced by waves wind and ocean currents the boundary controller is designed based on lyapunov s direct method and the backstepping technique proof of existence and uniqueness of the solutions of the closed loop control system is carried out by using the galerkin approximation method simulation results illustrate the effectiveness of the proposed boundary controller

interfacial dynamics for stokes flow theoretical and computational aspects of the method of interfacial dynamics for flow at vanishing reynolds number are considered the mathematical formulation relies on the boundary integral representation that expresses the flow in terms of distributions of stokes flow singularities over the interfaces the densities of the distributions are identified with the jump in the hydrodynamic traction due to interfacial in plane and transverse tensions the interfacial velocity or the strength of a hydrodynamic potential the numerical procedure involves describing the interfaces in terms of interfacial marker points that reproduce the evolving shapes of the interfaces by global or local interpolation solving integral equations of the second kind for the interfacial velocity or for the density of a hydrodynamic potential and computing the motion of the marker points while simultaneously updating interfacial fields relevant to the dynamics including the concentration of a surfactant and the position of interfacial point particles at an equilibrium configuration interfaces exhibiting isotropic tension elastic tensions viscous and incompressible behavior are considered the mathematical modeling of the tensions and bending moments developing over interfaces with a membrane like constitution is discussed in the context of the theory of thin shells to facilitate the numerical implementation the coupling of the interfacial mechanics to the hydrodynamics by means of interface force and torque balances is formulated in global cartesian coordinates recent progress in the implementation of boundary element methods is reviewed and areas for further research are identified

an adaptation of the fast multipole method for evaluating layer potentials in two dimensions abstract standard implementations of the fast multipole method which compute fields due to point sources or dipoles cannot be used to accurately evaluate the single and double layer potentials of potential theory close to the boundary or on the boundary when the boundary curves back on itself we describe the modifications necessary to accurately evaluate layer potentials in two dimensions which include quadrature rules for the short range contributions to the field continuous multipole moments for long range contributions and a more complex bookkeeping procedure we give formulae for second third and fourth order methods we show tests to verify the correctness of the method and numerical results which demonstrate the usefulness of the method for evaluating layer potentials near the boundary

modeling and boundary control of translational and rotational motions of nonlinear slender beams in three dimensional space abstract equations of motion of extensible and shearable slender beams with large translational and rotational motions under external loads in three dimensional space are first derived in a vector form boundary feedback controllers are then designed to ensure that the beams are practically k exponentially stable at the equilibrium the control design well posedness and stability analysis are based on two lyapunov type theorems developed for a class of evolution systems in hilbert space numerical simulations on a slender beam immersed in sea water are included to illustrate the effectiveness of the proposed control design

fourier analysis of stabilized q 1 q 1 mixed finite element approximation we use fourier analysis to investigate the instability of an equal order mixed finite element approximation method for elliptic incompressible flow equations the lack of stability can be attributed to the fact that the associated discrete ladyzhenskaya babuska brezzi lbb constant tends to zero as the mesh size is reduced we develop a stabilization approach that is appropriate to the periodic setting and deduce optimal choices of the associated stabilization parameter