discrete boundary conditions for problems with interface a discrete boundary condition on an artificial boundary is obtained for some second order elliptic boundary value problems with interface the problem is reduced to a problem without singularities and the standard finite element method is used to solve this problem some numerical examples are given

boundary methods for solving elliptic problems with singularities and interfaces boundary approximation techniques are described for solving homogeneous self adjoint elliptic equations piecewise expansions into particular solutions are used which approximate both the boundary and interface conditions in a least squares sense convergence of such approximations is proved and error estimates are derived in a natural norm numerical experiments are reported for the singular motz problem which yield extremely accurate solutions with only a modest computational effort

an investigation concerning optimal design of solid elastic plates abstract we consider the problem of maximizing the integral stiffness of solid elastic plates described by thin plate theory assuming the material volume and plate domain to be given we use the plate thickness function as the design variable and take both maximum and minimum allowable thickness values into account on the basis of a convenient tensorial formulation of the problem where the governing equations are derived by variational analysis and constitute necessary conditions for stationarity we develop an efficient and quite general numerical algorithm by means of which a number of stationary solutions for rectangular and axisymmetric annular plates with various boundary conditions are obtained these numerical results enable us to investigate the optimization problem itself in terms of its major parameters particularly the maximum and minimum values specified for the plate thickness for problems associated with large ratios between these constraint values plate designs with significant integral stiffeners are obtained we find however that these designs are only local optima and that a global optimal plate thickness function does generally neither exist within the class of smooth functions nor within the class of smooth functions with a finite number of discontinuities in order to determine a global optimal solution associated with given thickness constraint values it is therefore necessary to change the optimal design formulation with implications for a number of similar two dimensional optimization problems our results offer valuable indications of the lines along which such changes should be performed

on the use of consistent approximations for the optimal design of beams this paper presents a discretization strategy based on the concept of consistent approximations for certain optimal beam design problems where the beam is modeled using euler bernoulli beam theory it is shown that any accumulation point of the sequence of the stationary points of the family of resulting approximating problems is a stationary point of the original infinite dimensional problem the construction of approximating problems requires the development of a relaxation of constraints to ensure existence of solutions the numerical solution of the approximating problems by means of nonlinear programming algorithms that are not scale invariant must be preceded by a change of variables to guard against deterioration of performance the use of such approximating problems in conjunction with a diagonalization strategy is illustrated by a numerical example

refined and microlocal kakeya nikodym bounds of eigenfunctions in higher dimensions we prove a kakeya nikodym bound on eigenfunctions and quasimodes which sharpens a result of the authors blair and sogge in anal pde 8 747 764 2015 and extends it to higher dimensions as in the prior work the key intermediate step is to prove a microlocal version of these estimates which involves a phase space decomposition of these modes that is essentially invariant under the bicharacteristic geodesic flow in a companion paper blair and sogge in j differ geom 2015 it will be seen that these sharpened estimates yield improved l q m bounds on eigenfunctions in the presence of nonpositive curvature when 2 q frac 2 d 1 d 1

a review of an optimal design problem for a plate of variable thickness we revisit a classic design problem for a plate of variable thickness under the model of kirchhoff our main contribution has two goals one is to provide a rather general existence result under a main assumption on the structure of the tensor of material constants the other focuses on providing a minimal number of additional design variables for a relaxation of the problem when that assumption on the tensor of elastic constants does not hold in both situations the cost functional can be pretty general

a finite element method for laplace and helmholtz type boundary value problems with singularities laplace and helmholtz type boundary value problems with singularities are considered a sequence of approximations to the exact boundary conditions at an artificial boundary is given then the original problem is reduced to a boundary value problem in a domain away from the singularities furthermore finite element approximations are applied to this problem and error estimates are obtained finally some numerical examples show the effectiveness of this method

high order boundary conditions and finite elements for infinite domains a finite element method for the solution of linear elliptic problems in infinite domains is proposed the two dimensional laplace helmholtz and modified helmholtz equations outside an obstacle and in a semi infinite strip are considered in detail in the proposed method an artificial boundary b is first introduced to make the computational domain finite then the exact nonlocal dirichlet to neumann dtn boundary condition is derived on b this condition is localized and a sequence of local boundary conditions on b of increasing order is obtained the problem in with a localized dtn boundary condition on b is then solved using the finite element method the numerical stability of the scheme is discussed a hierarchy of special conforming finite elements is developed and used in the layer adjacent to b in conjunction with the local high order boundary condition applied on b an error analysis is given for both nonlocal and local boundary conditions numerical experiments are presented to demonstrate the performance of the method

an efficient finite element method for treating singularities in laplace s equation we present a new finite element method for solving partial differential equations with singularities caused by abrupt changes in boundary conditions or sudden changes in boundary shape terms from the local solution supplement the ordinary basis functions in the finite element solution all singular contributions reduce to boundary integrals after a double application of the divergence theorem to the galerkin integrals and the essential boundary conditions are weakly enforced using lagrange multipliers the proposed method eliminates the need for high order integration improves the overall accuracy and yields very accurate estimates for the singular coefftcients it also accelerates the convergence with regular mesh refinement and converges rapidly with the number of singular functions although here we solve the laplace equation in two dimensions the method is applicable to a more general class of problems

structural optimization of thin elastic plates the three dimensional approach the natural way to find the most compliant design of an elastic plate is to consider the three dimensional elastic structures which minimize the work of the loading term and pass to the limit when the thickness of the design region tends to zero in this paper we study the asymptotics of such a compliance problem imposing that the volume fraction remains fixed no additional topological constraint is assumed on the admissible configurations we determine the limit problem in different equivalent formulations and we provide a system of necessary and sufficient optimality conditions these results were announced in bouchitte et al c r acad sci paris ser i 345 713 718 2007 furthermore we investigate the vanishing volume fraction limit which turns out to be consistent with the results in bouchitte and fragala arch rat mech anal 184 257 284 2007 siam j control optim 46 1664 1682 2007 finally some explicit computation of optimal plates are given

riemannian manifolds with maximal eigenfunction growth on any compact riemannian manifold m g of dimension n the l sp 2 normalized eigenfunctions phi sb lambda satisfy lvert rvert phi sb lambda lvert rvert sb infty leq c lambda sp n 1 2 where delta phi sb lambda lambda sp 2 phi sb lambda the bound is sharp in the class of all m g since it is obtained by zonal spherical harmonics on the standard n sphere s sp n but of course it is not sharp for many riemannian manifolds for example for flat tori mathbb r sp n gamma we say that s sp n but not mathbb r sp n gamma is a riemannian manifold with maximal eigenfunction growth the problem that motivates this paper is to determine the m g with maximal eigenfunction growth our main result is that such an m g must have a point x where the set mathscr l sb x of geodesic loops at x has positive measure in s sp ast sb xm we show that if m g is real analytic this puts topological restrictions on m for example only m s sp 2 or m mathbb r p sp 2 topologically in dimension 2 can possess a real analytic metric of maximal eigenfunction growth we further show that generic metrics on any m fail to have maximal eigenfunction growth in addition we construct an example of m g for which mathscr l sb x has positive measure for an open set of x but which does not have maximal eigenfunction growth thus it disproves a naive converse to the main result

concerning the lp norm of spectral clusters for second order elliptic operators on compact manifolds in this paper we study the lp l2 mapping properties of a spectral projection operator for riemannian manifolds this operator is a generalization of the harmonic projection operator for spherical harmonics on s see c d sogge duke math j 53 1986 43 65 among other things we generalize the l2 restriction theorems of c fefferman e m stein and p tomas bull amer math sot 81 1975 477478 for the fourier transform in r to the setting of riemannian manifolds we obtain these results for the spectral projection operator as a corollary of a certain sobolev inequality involving d r2 for large r this sobolev inequality generalizes certain results for r of c kenig a ruiz and the author duke math j 55 1987 329 347 the main tools in the proof of the sobolev inequalities for riemannian manifolds are the hadamard parametrix cf l hbrmander acta mad 88 1968 341 370 and the analysis of linear partial differential equations vol iii springer verlag new york 1985 and oscillatory integral theorems of l carleson and p sjolin studia math 44 19x 287 299 and stein arm math stud 112 1986 307 357 0 1988 academic pnss iw

an elementary proof that the biharmonic green function of an eccentric ellipse changes sign p r garabedian showed in 1951 that the green function for the biharmonic boundary value problem with vanishing dirichlet data changes sign in case the domain is a sufficiently eccentric ellipse this refuted a conjecture made by j hadamard in 1908 the proof of garabedian was based on kernel functions the present note gives an elementary proof

optimization of clamped plates with discontinuous thickness we consider a new variational method for a clamped plate model and related shape optimization problems our approach allows the study of plates with discontinuous thickness

a finite element method for solving helmholtz type equations in waveguides and other unbounded domains a finite element method is described for solving helmholtz type boundary value problems in unbounded regions including those with infinite boundaries typical examples include the propagation of acoustic or electromagnetic waves in waveguides the radiation condition at infinity is based on separation of variables and differs from the classical sommerfeld radiation condition it is shown that the problem may be replaced by a boundary value problem on a fixed bounded domain the behavior of the solution near infinity is incorporated in a nonlocal boundary condition this problem is given a weak or variational formulation and the finite element method is then applied it is proved that optimal error estimates hold

linear and bilinear estimates for oscillatory integral operators related to restriction to hypersurfaces abstract we obtain bilinear estimates for oscillatory integral operators which are variable coefficient generalizations of bilinear restriction estimates for hypersurfaces as applications we improve the known estimates for oscillatory integrals

a duality approach in the optimization of beams and plates we introduce a class of nonlinear transformations called resizing rules which associate with optimal shape design problems certain equivalent distributed control problems while preserving the state of the system this puts into evidence the duality principle that the class of system states that can be achieved under a prescribed force via modifications of the structure shape of the system can be obtained as well via the modifications of the force action under a prescribed structure we apply such transformations to the optimization of beams and plates and in the simply supported or cantilevered cases the obtained control problems are even convex in all cases we establish existence theorems for optimal pairs by assuming only boundedness conditions moreover in the simply supported case we also prove the uniqueness of the global minimizer a general algorithm that iterates between the original transformed problems is introduced and studied the applications also include the case of variational inequalities

the singular finite element method for some elliptic boundary value problem with interface a kind of numerical method is proposed for some elliptic boundary value problems with interface it includes mainly two steps at the first step we obtain the approximation of the singularity near the singular points by solving a simple eigenvalue problem which is one dimension less than the original problem at the second step we apply the approximation of the singularity together with the standard finite element basis functions to construct the singular finite element space and finally solve the original problem on a conventional mesh some numerical examples show the effectiveness of this method

a finite element method for stokes equation using discrete singularity expansion the numerical method presented in this paper for stokes equation with corner singularity includes mainly two steps firstly we solve a simple eigenvalue problem which is one dimension less than the original problem to obtain the discrete expansion of the singularity near the corner secondly we combine the approximation of the singularity and standard finite element basis functions to construct special finite element spaces and solve the original problem in the special spaces on a conventional mesh the numerical examples show the effectiveness of this method

existence for shape optimization problems in arbitrary dimension we discuss some existence results for optimal design problems governed by second order elliptic equations with the homogeneous neumann boundary conditions or with the interior transmission conditions we show that our continuity hypotheses for the unknown boundaries yield the compactness of the associated characteristic functions which in turn guarantees convergence of any minimizing sequences for the first problem in the second case weaker assumptions of measurability type are shown to be sufficient for the existence of the optimal material distribution we impose no restriction on the dimension of the underlying euclidean space

discrete boundary conditions for elasticity problems with singularities discrete boundary conditions on an artificial boundary are obtained for the elasticity problems with singularities an iteration method is designed to solve the boundary value problem by using the discrete boundary conditions some numerical results are given which demonstrate the effectiveness of the numerical method

a finite element method for domains with corners a new finite element method is devised for the numerical solution of elliptic boundary value problems with geometrical singularities in it the singularity is eliminated form the computational domain in an exact fashion this is in contrast to other common methods such as those which use a refined mesh in the singularity region or those which use special singular finite elements in them the singularity is treated as a part of the numerical scheme the new method is illustrated on an elliptic differential equation in a domain containing a re entrant corner numerical experiments show that the new method yields result which are generally much more accurate than those obtained by using the standard finite element method with mesh refinement in the singularity region both methods require about the same computing time

optimality conditions for mass design problems and applications to thin plates we derive necessary and sufficient optimality conditions for a quite large class of structural design problems which can be formulated as follows under a given load and a total volume constraint minimize a suitable notion of compliance among all admissible mass distributions represented by positive measures with prescribed integral mean as a special case we focus attention on the optimization of thin plates we detail the corresponding optimality conditions and we show how they can be handled in order to determine analytically some optimal plates