computational formulation for periodic vibration of geometrically nonlinear structures part 2 numerical strategy and examples abstract in this paper the numerical strategy for solving the matrix amplitude equation with parameter is discussed in detail the matrix amplitude equation is the result of application of the galerkin method for analysis of periodic solutions of the geometrically nonlinear structures a theoretical background of the proposed method of analysis is given in a companion paper by lewandowski lewandowski r 1996 computational formulation for periodic vibration of geometrically nonlinear structures part 1 theoretical background international journal of solids and structures34 15 1925 1947 an example application of general theory is also presented numerical results are given and discussed to show some interesting behaviors of nonlinear structures and to illustrate the numerical efficiency and validity of the suggested method

finite element method for nonlinear free vibrations of composite plates a multimode time domain modal formulation based on the finite element method for large amplitude free vibration of thin composite plates is presented accurate frequency maximum deflection relations can be predicted for the fundamental and the higher nonlinear modes a modal participation is defined and accurate and convergent frequencies can be determined with minimum number of linear modes a procedure for the selection of initial conditions for periodic plate response is presented convergence of frequency with gridwork refinement and number of linear modes is studied the classical single mode elliptic function frequency solutions for simply supported beams and square plates are assessed examples of orthotropic and composite plates are given and the characteristics of nonlinear response are studied

computing puiseux series solutions to determinantal equations via combinatorial relaxation let a t x a ij t x be a square matrix with a ij being a polynomial in t and x this paper proposes an algorithm for computing the puiseux fractional power series solutions x x t to the equation operatorname det a t x 0 the algorithm is based on an observation which links the newton diagram polygon for det operatorname det a t x with the perfect matchings of a bipartite graph associated with a the algorithm is efficient making full use of available fast network type algorithms

asymmetric non linear forced vibrations of free edge circular plates part 1 theory in this article a detailed study of the forced asymmetric non linear vibrations of circular plates with a free edge is presented the dynamic analogue of the von karman equations is used to establish the governing equations the plate displacement at a given point is expanded on the linear natural modes the forcing is harmonic with a frequency close to the natural frequency omega n of one asymmetric mode of the plate thus the vibration is governed by the two degenerated modes corresponding to omega n which are one to one internally resonant an approximate analytical solution using the method of multiple scales is presented attention is focused on the case where one configuration which is not directly excited by the load gets energy through non linear coupling with the other configuration slight imperfections of the plate are taken into account experimental validations are presented in the second part of this paper

on matrix perturbations with minimal leading jordan structure we show that any matrix perturbation of an n n nilpotent complex matrix is similar to a matrix perturbation whose leading coefficient has minimal jordan structure additionally we derive the property that for matrix perturbations with minimal leading jordan structure the sufficient conditions of lidskii s perturbation theorem for eigenvalues are necessary too it is further shown how minimality can be obtained by computing a similarity transform whose entries are polynomials of degree at most n this relies on an extension of both lidskii s theorem and its newton diagram based interpretation

non linear steady state vibration of frames by finite element method a finite element method based on the virtual work principle to determine the steady state response of frams in free or forced periodic vibration is introduced the axial and flexural deformations are coupled by mean of the induced axial force along the element the spatial discretization of the deformations is achieved by the usual finite element method and the time discretization by fourier coefficients of the nodal displacements no unconventional element matrices are needed after applying the harmonic balance method a set of non linear algebraic equations of the fourier coefficients is obtained these equations are solved by the newtonian iteration method in terms of the fourier coefficient increments nodal damping can easily be included by a diagonal damping matrix the direct numerical determination of the fourier coefficient increments is difficult owing to the presence of peaks loops and discontinuities of slope along the amplitude frequency response curves parametric construction of the response curves using the phase difference between the response and excitation is recommended to provide more points during the rapid change of the phase i e at resonance for undamped natural vibration the method of selective coefficients adopted numerical examples on the duffing equation a hinged hinged beam a clamped hinged beam a ring and a frame are given for reasonably accurate results it is shown that the number of finite elements must be sufficient to predict at least the linear mode at the frequency of interest and the number of harmones considered must satisfy the conditions of completeness and balanceability which are discussed in detail

geometrical non linear steady state forced periodic vibration of plates part ii stability study and analysis of multi modal response the steady state geometrically non linear periodic vibration of rectangular thin plates under harmonic external excitations is analyzed using the hierarchical finite element and the harmonic balance methods modal coupling due to internal resonance is detected and the consequent multi modal and multi frequency response is demonstrated the stability of the obtained solutions is investigated by studying the evolution of perturbations to the solutions and using floquet s theory

comments on the method of harmonic balance the advantages and limitations of the harmonic balance or describing function approximation scheme for solving nonlinear ordinary differential equations of oscillatory motion are discussed advantages include appicability to equations of any order and with large degrees of nonlinearity ease of determining limit cycle behavior and its stability and overall speed and efficiency the limitation rules are essentially those described by mickens 1983 it is pointed out that perturbation procedures provide better results when the degree of nonlinearity is small

theory and experiments for nonlinear vibrations of imperfect rectangular plates with free edges abstract large amplitude vibrations of completely free rectangular plates are investigated for the first time nonlinear higher order shear deformation theory is used and the nonlinear response to transverse harmonic excitation in the frequency neighborhood of the fundamental mode is studied geometric imperfections are taken into account the numerical analysis is carried out in two steps first the plate displacements and rotations are expanded in terms of chebyshev polynomials and a linear analysis is conducted to obtain the natural frequencies and modes of vibration then the energy functional is discretized by using the natural modes of vibration and a system of nonlinear ordinary differential equations with cubic and quadratic nonlinear terms is obtained a pseudo arc length continuation and collocation scheme is used for bifurcation analysis of periodic solutions moreover to validate the numerical results experimental tests have been conducted for the fundamental vibration mode at several harmonic excitation amplitudes by using a laser doppler vibrometer to characterize the nonlinear response of the plate with imperfections and closed loop control of the excitation the effects of geometric imperfections on the trend of nonlinearity and on natural frequencies are fully discussed and the convergence of the numerical solutions has been verified

geometrical non linear steady state forced periodic vibration of plates part i model and convergence studies a model for the steady state geometrically non linear periodic vibration of thin rectangular plates under harmonic external excitation is presented the equations of motion in the time domain are derived by applying the principle of virtual work and the hierarchical finite element method hfem these equations are transformed into the frequency domain by the harmonic balance method hbm and are solved by a continuation method the convergence properties of the model are discussed by applying it to isotropic and to composite laminated plates

nonuniversality in low volume fraction ostwald ripening we study ostwald ripening in the regime of small volume fraction and consider spatially periodic systems whose size is smaller than the screening length within the snapshot perspective we obtain an explicit characterization of the leading order deviation to the classical mean field theory by lifshitz slyozov and wagner lsw using this representation we show that the corrections are not universal in the sense that the mean value has a strong dependence on geometry and arbitrarily large fluctuations can happen with finite probability

nonlinear analytic semiflows in this paper a local existence and regularity theory is given for nonlinear parabolic initial value problems x t f x t and quasilinear initial value problems x t a x t x t f x t this theory extends the theory of daprato and grisvard of 1979 and shows how various properties like analyticity of solutions can be derived as a direct corollary of the existence theorem

multi modal geometrical non linear free vibration of fully clamped composite laminated plates abstract the geometrically non linear free vibration of thin composite laminated plates is investigated using the hierarchical finite element hfem and the harmonic balance methods hbm von karman s non linear strain displacement relationships are employed and the mid plane in plane displacements are included in the model the equations of motion are developed by applying the principle of virtual work and are solved by a continuation method the convergence properties of the hfem and of the hbm are analyzed internal resonances are discovered and the consequent multi modal and multi frequency vibration of the plates is shown the variation of the plates mode shape and the effect of the fibres orientation are investigated

some aspects of the dynamic of v h h abstract we consider the evolution of a surface t according to the equation v h h where v is the normal velocity of t h is the sum of the two principal curvatures and h is the average of h on t we study the case where t intersects orthogonally a fixed surface and discuss some aspects of the dynamics of t under the assumption that the volume of the region enclosed between t and is small we show that in this case if 0 is near a hemisphere t keeps its almost hemispherical shape and slides on crawling approximately along orbits of the tangential gradient h of the sum h of the two principal curvatures of we also show that if p is a nondegenerate zero of h and a 0 is sufficiently small then there is a surface of constant mean curvature which is near a hemisphere of radius a with center near p and intersects orthogonally

a symplectic galerkin method for non linear vibration of beams and plates abstract non linear free vibration of beams and plates is conservative in the sense that the total energy of the system remains constant symplectic numerical integration aims to preserve the energy momenta and area volume of the phase space and is ideal in the study of non linear free vibration in this paper the lagrangian of continuous systems is discretized by galerkin s method to obtain the discrete hamiltonian and hamilton s equations symplectic numerical integration schemes are applied to the resulting ordinary differential equations to construct the phase diagram it is shown that due to modal coupling bounded quasi periodic and non periodic free vibrations are very common for continuous systems comparison is made with the runge kutta method it is shown that the runge kutta integration reduces the total energy of an undamped system while symplectic integration almost preserves it the symplectic schemes are extended to ordinary differential equations by means of the lagrangian and hamiltonian and the non linear vibration of beams and plates is studied by the schemes for the first time non linear modal coupling is emphasized the extension to shell structures involves the complication of curvature effects research is in progress

a center manifold analysis for the mullins sekerka model abstract the mullins sekerka model is a nonlocal evolution model for hypersurfaces which arises as a singular limit for the cahn hilliard equation we show that classical solutions exist globally and tend to spheres exponentially fast provided that they are close to a sphere initially our analysis is based on center manifold theory and on maximal regularity

the dynamics of nucleation for the cahn hilliard equation when a constant metastable solution of the cahn hilliard equation is subjected to a spatially localized large amplitude perturbation a transition process may be triggered leading to a globally stable stationary solution in one space dimension the existence and instability of a third stationary solution with the same mass is proved a spike like solution called a canonical nucleus within the class of solutions which are even with respect to the center of the spike it has a one dimensional unstable manifold in addition the process of nucleation by formal arguments using two space scales and two timescales is described the last stage in the process can be approximated by a nonlinear stefan free boundary problem

on the slowness of phase boundary motion in one space dimension we study the limiting behavior of the solution of u t e 2 u xx u 3 u 0 a x b with a neumann boundary condition or an appropriate dirichlet condition the analysis is based on energy methods we assume that the initial data has a transition layer structure i e ue 1 except near finitely many transition points we show that in the limit as e 0 the solution maintains its transition layer structure and the transition points move slower than any power of e

ostwald ripening for dilute systems under quasistationary dynamics we consider a dilute mixture in 3d of a finite number of particles initially close to spherical but of varying size evolving according to quasistationary dynamics under the scaling hypotheses that 1 size distance 2 deviation from sphericity size are both initially small we show robustness of the almost spherical shape by associating centers and radii to each particle and then derive rigorously a set of o d e s for the radii which we relate to the lifschitz slyozov wagner theory of coarsening and also establish an estimate for the speed of the centers

computational formulation for periodic vibration of geometrically nonlinear structures part 1 theoretical background abstract a general computational formulation for geometrically nonlinear structures excited by harmonic forces and executing periodic motion in a steady state is presented the equations of both continuous and discretized models are reformulated to obtain the motion equation in a more suitable form to a further analysis the multi harmonic solution of motion equation is written in a form of truncated fourier series next the galerkin ritz and the harmonic balance method are discussed in a context of their equivalency in derivation of the matrix amplitude equation the matrix amplitude equation as well as the associated tangent matrix are given in an explicit form the stability of steady state solution is discussed by using the floquet theory the numerical algorithm and an example application are described in a companion paper by lewandowski lewandowski r computational formulation for periodic vibration of geometrically nonlinear structures part 2 numerical strategy and numerical examples international journal of solids and structures in preparation

convergence of the cahn hilliard equation to the hele shaw model we prove that level surfaces of solutions to the cahn hilliard equation tend to solutions of the hele shaw problem under the assumption that classical solutions of the latter exist the method is based on a new matched asymptotic expansion for solutions a spectral analysis for linearizd operators and an estimate for the difference between the true solutions and certain approximate ones

derivation of the lsw theory for ostwald ripening by homogenization methods the coarsening of a solid phase in an undercooled liquid is described by a stefan problem with surface tension the solid phase is characterized by a large number of balls with small volume fraction and small capacity we solve the quasi static equation as well as the parabolic problem and construct approximate solutions by means of comparison principles the framework of young measures is used to pass to the limit of infinitely many particles we obtain that the capacity is the crucial quantity to get the conservation law for the particle size distribution which is studied in the classical theory for ostwald ripening by lifshitz slyozov wagner lsw

non linear free vibrations of beams by the finite element and continuation methods abstract a computational method is presented for determining backbone curves of arbitrary geometry as an example beam vibration is considered the beam response is expanded into a truncated fourier series with respect to time after starting from the integral of action the variational approach and the finite element method are used to formulate the non linear eigenvalue problem the continuation method is adopted to solve the resulting non linear eigenvalue problem and to obtain the non linear frequencies and modes of vibration numerical results for various beams are presented and compared with available results to demonstrate the accuracy and applicability of the method moreover the bifurcation points on some beams backbone curves are found and reported for the first time

nearly optimal algorithms for canonical matrix forms a las vegas type probabilistic algorithm is presented for finding the frobenius canonical form of an n times n matrix t over any field kk the algorithm requires softo mm n mm n cdot log n o 1 operations in kk where o mm n operations in kk are sufficient to multiply two n times n matrices over kk this nearly matches the lower bound of omega mm n operations in kk for this problem and improves on the o n 4 operations in kk required by the previously best known algorithms a fast parallel implementation of the algorithm is also demonstrated for the frobenius form which is processor efficient on a pram as an application we give an algorithm to evaluate a polynomial g in kk x at t which requires only softo mm n operations in kk when deg g leq n 2 other applications include sequential and parallel algorithms for computing the minimal and characteristic polynomials of a matrix the rational jordan form of a matrix for testing whether two matrices are similar and for matrix powering which are substantially faster than those previously known

nonlinear forced vibration of orthotropic rectangular plates using the method of multiple scales the method of multiple scales mms in conjunction with the galerkin method is used to analyze the nonlinear forced and damped response of a rectangular orthotropic plate subjected to a uniformly distributed harmonic transverse loading the effects of damping and in plane loads are considered the analysis considers simply supported as well as clamped plates for each case both movable and immovable edge conditions are considered by using mms all possible resonances such as primary subharmonic and superharmonic reso nances are studied for the undamped response without in plane loading comparisons of the mms results with those obtained by the finite element method show excellent agreement evelopment of composite materials comprising lam inates of orthotropic or multilayered anisotropic materi als recently has been receiving substantially growing research efforts due to the increasing demands for energy efficient high strength minimum weight aircraft designs many re searchers believe that the use of composite materials offers promising alternatives for aircraft designs thin laminated composite plates subjected to transverse periodic loadings could encounter deflections of the order of plate thickness or even higher responses of this kind cannot be predicted by using the linear theory consequently the need to study large amplitude deflection vibrations of composite structures is of paramount importance the literature survey shows that the equations of motion for the large deflection analysis of heterogeneous anisotropic plates using the von karman geometrical nonlinearity were first considered by whitney and leissa 1 based on these equa tions different methods of analysis have been developed by several researchers an excellent number of collections on nonlinear free and forced vibrations of composite plates cov ering the work through 1979 can be found in the comprehen sive book by chia 2 bert3 has conducted a survey on the dynamics of composite plates for the period 1979 81 a re view of the literature on nonlinear vibrations of plates can be found in the review paper by sathyamoorthy4 and the book by nayfeh and mook 5 large deflection analysis of symmetrically laminated rec equations of motion are presented in terms of the lateral displacement and stress function the equations are nondi mensionalized following the transformation introduced by brunelle and oyibo 9 though multimode analysis can be treated the present study is focused on single mode analysis a deflection function representing the first mode and satisfy ing the boundary conditions is assumed and subsequently the stress function is found next the galerkin method is applied to obtain the modal equation which is solved analytically by using the method of multiple scales mms 10 the mms also provides solutions for subharmonic and superharmonic reso nances the effects of damping ratio plate aspect ratio and in plane loading then are studied

calcul des invariants de malgrange et de gerard levelt d un systeme differentiel lineaire en un point singulier irregulier we consider a system of ordinary differential equations y x a x y x where a x is a matrix with a pole of the order q at x 0 for this system b malgrange has introduced the quantity i d which can be computed with the cyclic vector method the present paper treats the practical evaluation of i d and the invariants 1 2 q 1 of gerard and levelt without using a cyclic vector first we shall give a survey of the algorithm given in hilali and wazner numer math 50 1987 429 449 by which a system is reduced to a super irreducible form we show that these invariants follows directly from this form

thermal effects on geometrically nonlinear vibrations of rectangular plates with fixed edges nonlinear forced vibrations and postbuckling of isotropic rectangular plates subjected to thermal variations are studied geometric imperfections are taken into account since they play a fundamental role the plate is modelled by using the von karman hypothesis and the equations of motion are obtained by using an energy approach plates with fixed edges and additional rotational elastic constraint are considered a pseudo arclength continuation method is used in order to obtain numerical results laboratory experiments have been performed on two plates of different thickness

nonlinear vibrations of viscoelastic rectangular plates abstract nonlinear vibrations of viscoelastic thin rectangular plates subjected to normal harmonic excitation in the spectral neighborhood of the lowest resonances are investigated the von karman nonlinear strain displacement relationships are used and geometric imperfections are taken into account the material is modeled as a kelvin voigt viscoelastic solid by retaining all the nonlinear terms the discretized nonlinear equations of motion are studied by using the arclength continuation and collocation method numerical results are obtained for the fundamental mode of a simply supported square plate with immovable edges by using models with 16 and 22 degrees of freedom and investigating solution convergence comparison to viscous damping and the effect of neglecting nonlinear viscoelastic damping terms are shown the change of the frequency response with the retardation time parameter is also investigated as well as the effect of geometric imperfections

theory and experiments for large amplitude vibrations of rectangular plates with geometric imperfections abstract large amplitude vibrations of rectangular plates subjected to harmonic excitation are investigated the von karman nonlinear strain displacement relationships are used to describe the geometric nonlinearity a specific boundary condition with restrained normal displacement at the plate edges and fully free in plane displacements not previously considered has been introduced as a consequence that it is very close to the experimental boundary condition results for this boundary condition are compared to nonlinear results previously obtained for i simply supported plates with immovable edges ii simply supported plates with movable edges and iii fully clamped plates the nonlinear equations of motion are studied by using a code based on pseudo arclength continuation method a thin rectangular stainless steel plate has been inserted in a metal frame this constraint is approximated with good accuracy by the newly introduced boundary condition the plate inserted into the frame has been measured with a 3d laser system in order to reconstruct the actual geometry and identify geometric imperfections out of planarity the plate has been experimentally tested in laboratory for both the first and second vibration modes for several excitation magnitudes in order to characterize the nonlinearity of the plate with imperfections numerical results are able to follow experimental results with good accuracy for both vibration modes and for different excitation levels once the geometric imperfection is introduced in the model effects of geometric imperfections on the trend of nonlinearity and on natural frequencies are shown convergence of the solution with the number of generalized coordinates is numerically verified

on the steady state response and stability of non linear oscillators using harmonic balance abstract the harmonic balance hb method is used to study the steady state periodic response as well as its stability of a softening duffing oscillator the objectives of the work are 1 to show how the qualitative nature of the hb solution changes if one uses two harmonics rather than one in the approximate solution and 2 to show that the analysis of the mathieu equation which governs local stability of the steady state response must be based on an approximate rather than an exact analysis of this mathieu equation thus floquet theory incorrectly describes the stability of the approximate solution

large amplitude finite element flexural vibration of plates stiffened plates large amplitude free flexural vibration of stiffened and unstiffened plates has been studied by using the finite element method an isoparametric quadratic plate bending element has been used both for the plate and the stiffener the dynamic version of von karman s field equations has been adopted and the formulation has been done in the total lagrangian coordinate system the in plane deformation and inertia have been taken into account the resulting nonlinear equations have been solved by the direct iteration technique using a linear mode shape as the starting vector the stiffener has been elegantly modeled so that it can be placed anywhere within the plate element and it need not follow the nodal lines this has increased the flexibility of the mesh generation considerably the arbitrary orientation and eccentricity of the stiffener have been incorporated in the formulation the shear deformation has been incorporated according to mindlin s hypothesis stiffened and unstiffened plates that have various boundary conditions have been analyzed and the results have been compared with those available in the literature

the harmonic balance method for bifurcation analysis of large scale nonlinear mechanical systems abstract the harmonic balance hb method is widely used in the literature for analyzing the periodic solutions of nonlinear mechanical systems the objective of this paper is to exploit the method for bifurcation analysis i e for the detection and tracking of bifurcations of nonlinear systems to this end an algorithm that combines the computation of the floquet exponents with bordering techniques is developed a new procedure for the tracking of neimark sacker bifurcations that exploits the properties of eigenvalue derivatives is also proposed the hb method is demonstrated using numerical experiments of a spacecraft structure that possesses a nonlinear vibration isolation device

nonlinear dynamics of axially moving plates abstract the nonlinear dynamics for forced motions of an axially moving plate is numerically investigated using von karman plate theory and retaining in plane displacements and inertia the equations of motion are obtained via an energy method based on lagrange equations this yields a set of second order nonlinear ordinary differential equations with coupled terms the equations are transformed into a set of first order nonlinear ordinary differential equations and are solved via the pseudo arclength continuation technique the near resonance nonlinear dynamics is examined via plotting the frequency response curves of the system results are shown through frequency response curves time histories and phase plane diagrams the effect of system parameters such as the axial speed and the pretension on the resonant responses is also highlighted

nongeneric eigenvalue perturbations of jordan blocks abstract we show that if an n n jordan block is perturbed by an o e upper k hessenberg matrix k subdiagonals including the main diagonal then generically the eigenvalues split into p rings of size k and one of size r if r 0 where n pk r this generalizes the familiar result k n p 1 r 0 that generically the eigenvalues split into a ring of size n we compute the radii of the rings to first order and generalize the result in a number of directions involving multiple jordan blocks of the same size

the effects of large vibration amplitudes on the mode shapes and natural frequencies of thin elastic structures part iii fully clamped rectangular isotropic plates measurements of the mode shape amplitude dependence and the spatial distribution of harmonic distortion abstract in this work the amplitude dependence of the resonance frequency and the spatial distribution of the first and second harmonic components of the dynamic response of a fully clamped rectangular homogeneous plate have been investigated experimentally by using electrodynamic point excitation at the centre of the plate and a non contacting optical vibration transducer for response measurement the spatial distribution of the complex harmonic distortion has been investigated in order to reach high amplitudes of vibration under distributed loading a set of tests were also carried out with a progressive wave tunnel in which the plates tested were subjected to high level acoustic excitation at overall sound pressure levels up to 155 db the results of such experimental measurements are given and compared with theoretical predictions

front migration in the nonlinear cahn hilliard equation the method of matched asymptotic expansions is used to describe solutions of the nonlinear cahn hilliard equation for phase separation in n 1 space dimensions the expansion is formally valid when the thickness of internal transition layers is small compared with the distance separating layers and with their radii of curvature on the dominant slowest timescale the interface velocity is determined by the mean curvature of the interface by a non local relation which is identical to that in a well known quasi static model of solidification which exhibits a shape instability discovered by mullins sekerka j appl phys 34 323 329 1963 on a faster timescale the cahn hilliard equation regularizes a classic two phase stefan problem similarity solutions of the two phase stefan problem should describe boundary layers existence and uniqueness of such similarity solutions which admit metastable states is proved rigorously in an appendix

on the parabolic stefan problem for ostwald ripening with kinetic undercooling and inhomogeneous driving force abstract ostwald ripening is the coarsening phenomenon caused by the diffusion and solidification process which occurs in the last stage of a first order phase transformation the force that drives the system towards equilibrium is the gradient of the chemical potential that according to the gibbs thomson condition on the interface is proportional to its mean curvature a quantitative description of ostwald ripening has been developed by the lifschitz slyozov wagner lsw theory we extend the work of niethammer 2000 15 which deals with kinetic undercooling in the quasi static case to the parabolic setting with temporally inhomogeneous driving forces on the solid liquid interfaces by means of a priori estimates local and global existence results for the parabolic stefan problem we derive a first order approximation for the dynamical equations for the heat distribution and particle radii and then prove the convergence to a limiting description using a mean field equation

an efficient boundary integral method for the mullins sekerka problem we use a boundary integral technique to study the two space dimensional mullins sekerka free boundary problem which originates from a study of solidification and liquidation of materials of negligible specific heat this is an area preserving and curve shortening motion evolution equations for the free boundaries are derived in terms of the tangent angle and total arclength which makes a small scale decomposition possible and the fourier transform a powerful tool in numerical calculations with this formulation implicit schemes can be implemented to avoid the difficult numerical stiffness associated with explicit schemes we can compute solutions up to the time when there is a topological change i e when particles touch or break up our numerical results for systems of a single particle or multi particles provide some valuable information in the particle dynamics such as the circularization of each individual particle and the mass transfer between different particles during particle interactions

ostwald ripening in two dimensions the rigorous derivation of the equations from the mullins sekerka dynamics we consider a dilute mixture of a finite number of particles and we are interested in the coarsening of the spatial distribution in two space dimensions under mullins sekerka dynamics under the appropriate scaling hypotheses we associate radii and centers to each particle and derive equations for the whole evolution

numerical methods for thermally stressed shallow shell equations abstract we develop efficient and accurate numerical methods to solve a class of shallow shell problems of the von karman type the governing equations form a fourth order coupled system of nonlinear biharmonic equations for the transverse deflection and airy s stress function a second order finite difference discretization with three iterative methods picard newton and trust region dogleg is proposed for the numerical solution of the nonlinear pde system three simple boundary conditions and two application motivated mixed boundary conditions are considered along with the nonlinearity of the system boundary singularities that appear when mixed boundary conditions are specified are the main numerical challenges two approaches that use either a transition function or local corrections are developed to deal with these boundary singularities all the proposed numerical methods are validated using carefully designed numerical tests where expected orders of accuracy and rates of convergence are observed a rough run time performance comparison is also conducted to illustrate the efficiency of our methods as an application of the methods a snap through thermal buckling problem is considered the critical thermal loads of shell buckling with various boundary conditions are numerically calculated and snap through bifurcation curves are also obtained using our numerical methods together with a pseudo arclength continuation method our results are consistent with previous studies

the hele shaw problem and area preserving curve shortening motions we prove existence locally in time of a solution of the following hele shaw problem given a simply connected curve contained in a smooth bounded domain find the motion of the curve such that its normal velocity equals the jump of the normal derivatives of a function which is harmonic in the complement of the curve in and whose boundary value on the curve equals its curvature we show that this motion is a curve shortening motion which does not change the area of the region enclosed by the curve in case is the whole plane 2 we also show that if the initial curve is close to an equilibrium curve i e to a circle then there exists a global solution and the global solution tends to some circle exponentially fast as time tends to infinity

a finite element time domain modal formulation for large amplitude free vibrations of beams and plates abstract a finite element time domain modal fomulation is presented for the large amplitude free vibration of plates the procedure of deriving the non linear modal equations of motion is simple and general accurate frequency maximum deflection relations can be obtained for the fundamental and higher non linear modes the percentage of participation from each linear mode determines the number of modes to be included for accurate frequency results numerical examples on beams and square plates are given and compared with classical continuum analytical solutions a plate of complex planform and boundary conditions is also given to show the versatility of the presented modal formulation

the effect of the geometry of the particle distribution in ostwald ripening based on 1 we derive equations for the radii and the centers that we relate to the lifshitz slyozov wagner theory

on the effect of stochastic fluctuations in the dynamics of the lifshitz slyozov wagner model in this paper the dynamics of a system of spherical particles that fill a small volume fraction of the space and that evolves in a concentration field is discussed corrections to the lifshitz slyozov wagner lsw model that take into account the stochastic character of the problem are computed it is proved under suitable smallness assumptions for the volume fraction filled by the particles that the effect of these corrections does not modify much the dynamics of the self similar solutions of the lsw system of equations

on first order corrections to the lsw theory ii finite systems we consider first order corrections to the classical theory by lifshitz slyozov and wagner lsw for systems with a finite number of particles numerical simulations in v e fradkov etal phys rev e 53 3925 3932 1996 show a cross over in the scaling of the correction term from 1 3 to 1 2 is the volume fraction of particles when the system size becomes larger than the screening length we rigorously derive this cross over for the rate of change of the energy starting from the monopole approximation the proof exploits the fact that the rate of change of energy has a variational characterization

shear deformable versus classical theories for nonlinear vibrations of rectangular isotropic and laminated composite plates in the present study i the classical von karman theory ii the first order shear deformation theory and iii the higher order third order shear deformation theory are compared for studying the nonlinear forced vibrations of isotropic and laminate composite rectangular plates in particular the harmonic response in the frequency neighborhood of the fundamental mode of rectangular plates is investigated and the response curves computed by using the three different theories are compared the boundary conditions of the plates are simply supported with immovable edges geometric imperfections are taken into account calculations for isotropic and laminated composite plates are presented and results are discussed for isotropic plates the frequency response curves for large amplitude vibrations obtained by using the three theories are almost coincident for laminated composite plates differences arise for relatively thick plates ratio between the thickness and the edge equal to 0 1 while for thin plates ratio between the thickness and the edge equal to 0 01 no difference is obtained for all cases the first order shear deformation with shear correction factor 3 2 and the higher order shear deformation theories give practically coincident results and differences are observed with respect to the classical von karman theory

the equations of ostwald ripening for dilute systems we consider a dilute mixture in 3d of a finite number of particles initially close to spherical but of varying sizes and representing one of the phases evolving according to the quasistatic dynamics under the scaling hypotheses that 1 typical size typical distance and 2 deviation from sphericity typical size are small we associate centers and radii to each particle for the whole evolution and derive rigorously a set of odes fo the radii which we relate to the lifschitz slyosov wagner theory of coarsening

screening in interacting particle systems we consider the green s function of the laplace operator in domains with spherical holes particles under natural assumptions on the distribution of particles we show that the green s function decays exponentially over distances larger than the screening length this result is fundamental for example when deriving effective equations for coarsening systems in unbounded domains