Determination of the flexural rigidity of a beam from limited boundary measurements

Inverse coefficient identification problems associated with the fourth-order Sturm-Liouville operator in the steady state Euler-Bernoulli beam equation are investigated. Unlike previous studies in which spectral data are used as additional information, in this paper only boundary information is used, hence non-destructive tests can be employed in practical applications.     

Generalized solutions for the Euler-Bernoulli model with distributional forces

We establish existence and uniqueness of generalized solutions to the initial-boundary value problem corresponding to an Euler-Bernoulli beam model from mechanics. The governing partial differential equation is of order four and involves discontinuous, and even distributional, coefficients and right-hand side. The general problem is solved by application of functional analytic techniques to obtain estimates for the solutions to regularized problems. Finally, we prove coherence properties and provide a regularity analysis of the generalized solution.     

Inverse conductivity Problem with Internal Data

This paper concerns the reconstruction of a scalar coefficient of a second-order elliptic equation in divergence form posed on a bounded domain from internal data. This problem finds applications in multi-wave imaging, greedy methods to approximate parameter-dependent elliptic problems, and image treatment with partial differential equations. We first show that the inverse problem for smooth coefficients can be rewritten as a linear transport equation. Assuming that the coefficient is known near the boundary, we study the well-posedness of associated transport equation as well as its numerical resolution using discontinuous Galerkin method. We propose a regularized transport equation that allow us to derive rigorous convergence rates of the numerical method in terms of the order of the polynomial approximation as well as the regularization parameter. We finally provide numerical examples for the inversion assuming a lower regularity of the coefficient, and using synthetic data.     

Coefficient identification in Euler-Bernoulli equation from over-posed data

A method for solving the inverse problem for coefficient identification in the Euler-Bernoulli equation from over-posed data is presented. The original inverse problem is replaced by a minimization problem. The method is applied to the problem for identifying the coefficient in the case when it is a piece-wise polynomial function. Several examples are elaborated and the numerical results confirm that the solution of the imbedding problem coincides with the direct simulation of the original problem within the second order of approximation.     

Analytical solution of a spatially variable coefficient advection–diffusion equation in up to three dimensions

Analytical solutions are provided for the two- and three-dimensional advection–diffusion equation with spatially variable velocity and diffusion coefficients. We assume that the velocity component is proportional to the distance and that the diffusion coefficient is proportional to the square of the corresponding velocity component. There is a simple transformation which reduces the spatially variable equation to a constant coefficient problem for which there are available a large number of known analytical solutions for general initial and boundary conditions. These solutions are also solutions to the spatially variable advection–diffusion equation. The special form of the spatial coefficients has practical relevance and for divergent free flow represent corner or straining flow. Unlike many other analytical solutions, we use the transformation to obtain solutions of the spatially variable coefficient advection–diffusion equation in two and three dimensions. The analytical solutions, which are simple to evaluate, can be used to validate numerical models for solving the advection–diffusion equation with spatially variable coefficients. For numerical schemes which cannot handle flow stagnation points, we provide analytical solution to the spatially variable coefficient advection–diffusion equation for two-dimensional corner flow which contains an impermeable flow boundary. The impermeable flow boundary coincides with a streamline along which the fluid velocity is finite but the concentration vanishes. This example is useful for validating numerical schemes designed to predict transport around a curved boundary.     

Solving Inverse Problems for Hyperbolic Equations via the Regularization Method

In the paper, we first deduce an optimization problem from an inverse problem for a general operator equation and prove that the optimization problem possesses a unique, stable solution that converges to the solution of the original inverse problem, if it exists, as a regularization factor goes to zero. Secondly, we apply the above results to an inverse problem determining the spatially varying coefficients of a second order hyperbolic equation and obtain a necessary condition, which can be used to get an approximate solution to the inverse problem.     

Nonlinear analysis of a parametrically excited beam with intermediate support by using Multi-dimensional incremental harmonic balance method

In this paper, a nonlinear Euler-Bernoulli beam under a concentrated harmonic excitation with intermediate nonlinear support is investigated. Continuous expression for the kinetic energy, potential energy and dissipation function are constructed. An energy method based on the Lagrange equation combined with the Galerkin truncation is used for discretizing the governing equation. The Multi-dimensional incremental harmonic balance method (MIHBM) is derived, and the comparisons between the numerical results and the approximate analytical solutions based on the MIHBM verify the excellent accuracy of the MIHBM. The steady state dynamic of the beam is investigated by MIHBM. In order to investigate the energy transmission and understand the vibration response of the Euler-Bernoulli beam, the effects of the key parameters on the dynamic behaviors are studied and discussed, individually. The results show that the amplitude-frequency curves exhibits softening nonlinear behavior in the super-harmonic resonance region, and near resonant region the hardening nonlinear behavior is observed depending on the different parameters. Nonlinear dynamic analysis, such as bifurcation, 3-D frequency spectrum, waveform, frequency spectrum, phase diagram and Poincaré map, are also presented in order to study the influences of the key parameters on the vibration behaviors for the beam in a more accurate manner. In addition, the path to chaotic motion is observed to be through a sequence of the periodic motion and quasi-periodic motion.     

Boundary Controllability and Observability of a One-Dimensional Nonuniform SCOLE System

A hybrid system, composed of an elastic beam governed by an Euler-Bernoulli beam equation with variable coefficients and a linked rigid body governed by an ordinary differential equation, is considered. Various controllability/observability properties of the system under bounday control/observation are studied. It is shown that an open-loop smooth/singular controller of either torque control or force control is sufficient to make the system exactly controllable in arbitrarily short time duration. The work was carried out with the support of the National Natural Science Foundation of China and the Russian Foundation for Fundamental Researches, Grant 02-01-00554.     

Mathematical model for the bending of plastically anisotropic beams

A mathematical model for the bending of a plastically anisotropic beam simply supported at both ends and subjected to a constant moment is considered. A differential equation with variable coefficients is derived for the beam curvature. The yield points of the beam material under tension and compression are assumed to be known. The elastoplastic bending of the beam with allowance for the strength-different (SD) effect is considered. The classical Bernoulli–Euler beam theory and the ideal plasticity model are used construct the mathematical model, and the problem is solved analytically. The solutions obtained for a classical isotropic beam and an SD beam are compared, and the contribution of the SD effect is analyzed. The problem is solved completely, and its results can be used to study bending under different loading.     

Analysis of coefficient identification problems associated to the inverse Euler-Bernoulli beam theory

A rigorous investigation of the identification of a heterogeneousflexural rigidity coefficient in the Euler-Bernoulli steady-statebeam theory in the presence of a prescribed load is presented.Mathematically, this study is an extension to higher-order differentialequations of the coefficient identification problem analysedby Marcellini (1982) for the one-dimensional Poisson equation.In addition, various types of boundary conditions are discussed.Conditions for the well-posedness of these inverse problemsare established and, furthermore, numerical results obtainedusing a regularization algorithm are presented.     

Numerical method for solving an inverse problem for a population model

The inverse problem of determining the growth rate coefficient of biological objects from additional information on their time-dependent density is considered. Two nonlinear integral equations are derived for the unknown coefficient, which is determined on part of its domain from one equation and on the remaining part from the other equation. The nonlinear integral equations are solved by iterative methods. The convergence conditions for the iterative methods are formulated, and results of numerical experiments are presented.     

A 2-component Camassa-Holm Equation,Euler-Bernoulli Beam Problem,and Noncommutative Continued Fractions

A new approach to the Euler-Bernoulli beam based on an inhomogeneous matrix string problem is presented. Three ramifications of the approach are developed:

1. motivated by an analogy with the Camassa-Holm equation a class of isospectral deformations of the beam problem is formulated;
2. a reformulation of the matrix string problem in terms of a certain compact operator is used to obtain basic spectral properties of the inhomogeneous matrix string problem with Dirichlet boundary conditions;
3. the inverse problem is solved for the special case of a discrete Euler-Bernoulli beam. The solution involves a noncommutative generalization of Stieltjes’ continued fractions, leading to the inverse formulas expressed in terms of ratios of Hankel-like determinants.

© 2022 Courant Institute of Mathematics and Wiley Periodicals LLC.     

A population model with time-dependent delay

We present analytical and computational results concerning the linear stability and instability of the uniform steady-state solution of a system of reaction-diffusion equations where a parameter in the kinetic terms is periodic in time. Under suitable assumptions the system is equivalent to a scalar equation with a periodically varying delay. Such a varying delay can model the seasonal fluctuations to the regeneration time of a resource. We study the effect such a varying delay can have on the stability of the spatially uniform steady-state. Analytical results reveal that instability can set in if the delays are large, while computational methods of analysing the stability equations reveal the precise shape of the instability boundary. The nonlinear stability of the uniform state is also examined using ladder methods.     

Modelling and bifurcation analysis of internal cantilever beam system on a steadily rotating ring

Using Hamilton variation principle, a nonlinear dynamic model of the system with a finite deforming Rayleigh beam clamped radially to the interior of a rotating rigid ring, under the assumption that the constitutive relation of the beam is linearly elastic, is discussed. The bifurcation behavior of the simple system with the Euler-Bernoulli beam is also discussed. It is revealed that these two models have no influence on the critical bifurcation value and buckling solution in the steady state. Then we use the assumption model method to analyse the bifurcation behavior of the steadily rotating Euler-Bernoulli beam and get two different types of bifurcation behavior which physically exist. Finite element method and shooting method are used to verify the analytical results. The numerical results confirm our research conclusion. Project supported by the National Natural Science Foundation of China (Grant No. 19332022) and Space High Technology Foundation of China.     

Exact solutions for coupled analysis of thin-walled functionally graded beams with non-symmetric single- and double-cells

In the present work, the exact solutions for coupled analysis for bending and torsional case thin-walled functionally graded (FG) beams with non-symmetric single- and double-cells are presented for the first time. For this purpose, an accurate and efficient method is proposed to obtain the FG member stiffness matrix based on the series expansions of displacement components. Three types of material distributions are considered and the beam mechanical properties are graded along the wall thickness according to a power law of the volume fraction. The present beam model is on the basis of the Euler-Bernoulli beam theory and the Vlasov one for bending and torsional problems, respectively. The explicit expressions for displacement parameters are derived using the power series approach from the four coupled equilibrium equations. Finally, the FG member stiffness matrix is determined from the seven force-displacement relations. In order to show the accuracy and super convergence of the thin-walled FG beam element developed by this study, the numerical solutions are presented and compared with results obtained from the finite beam element based on the approximate interpolation polynomials and other available results. Especially, the effects of various structural parameters such as material distribution type, volume fraction index, boundary condition, and material ratio on the spatially coupled responses of FG box beams with non-symmetric single- and double-cells are parametrically investigated.     

On a structural acoustic model which incorporates shear and thermal effects in the structural component

In this paper we consider a structural acoustic model which takes account of thermal effects over and above displacement, rotational inertia and shear effects in the flat flexible structural component of the model. Thus the structural medium is a Reissner-Mindlin plate into which an additional degree of freedom, viz. temperature variation in the plate, has been introduced and the constitutive equations for the structural acoustic model couple parabolic dynamics with hyperbolic dynamics. We show unique solvability of the mathematical model and investigate the effect of the presence of thermal effects on the mechanical dissipation devices needed to attain uniform stabilization of the two-dimensional model in which the structural component is a Timoshenko beam. It turns out that, as in linear structural acoustic models which use the Euler-Bernoulli equation or the Kirchoff equation to describe the deflections of the thermo-elastic structural medium, uniform stabilization of the energy associated with the model can be attained without introducing mechanical dissipation at the free edge of the beam. Open problems with regard to the stabilization of the three-dimensional model are outlined.     

A numerical method for reconstructing the coefficient in a wave equation

We present a numerical method for reconstructing the coefficient in a wave equation from a single measurement of partial Dirichlet boundary data. The original inverse problem is converted to a nonlinear integral differential equation, which is solved by an iterative method. At each iteration, one linear second‐order elliptic problem is solved to update the reconstruction of the coefficient, then the reconstructed coefficient is used to solve the forward problem to obtain the new data for the next iteration. The initial guess of the iterative method is provided by an approximate model. This model extends the approximate globally convergent method proposed by Beilina and Klibanov, which has been well developed for the determination of the coefficient in a special wave equation. Numerical experiments are presented to demonstrate the stability and robustness of the proposed method with noisy data.© 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 289–307, 2015     

A Direct Method for Solving the Inverse Coefficient Problem for an Elliptic Equation with Piecewise Constant Coefficients

Journal of Applied and Industrial Mathematics - Some direct numerical method is presented for solving the inverse coefficient problem for an elliptic equation with piecewise constant coefficients....     

Lipschitz stability in determining density and two Lamé coefficients

We consider an inverse problem of determining spatially varying density and two Lamé coefficients in a non-stationary isotropic elastic equation by a single measurement of data on the whole lateral boundary. We prove the Lipschitz stability provided that initial data are suitably chosen. The proof is based on a Carleman estimate which can be obtained by the decomposition of the Lamé system into the rotation and the divergence components.     

On an exact numerical simulation of solitary-wave solutions of the Burgers–Huxley equation through Cardano’s method

In this work, we develop an exact finite-difference methodology to approximate the solution of a diffusive partial differential equation with Burgers advection and Huxley reaction law. The model under investigation possesses solitary-wave solutions which are positive, bounded, and both spatially and temporally monotone. On the other hand, our computational model is a nonlinear technique for which the new approximations are provided as the roots of an uncoupled system of cubic polynomials, in which the constant coefficients are functions of the model parameters and the numerical step-sizes. In this system, each cubic equation is solved using Cardano’s formulas. The method proposed in this work preserves the positivity, the boundedness and the monotonicity of approximations, as well as the constant solutions of the continuous model. The simulations provided in this work show a good agreement with respect to the analytical solutions employed.