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The linearity assumption is widely used when acquiring the hydrodynamic coefficients of a floating structure. However, the linear damping is frequently underestimated, especially for the natural frequency. To investigate the sloping seafloor effects on the damping terms of a single module of a semi-submersible Very Large Floating Structure (VLFS), this paper revisits the conventional formulation and further proposes the direct integration method for obtaining the linear and quadratic damping coefficients from free-decay tests. Numerical free-decay simulations of the single module over variable bathymetry are carried out by the CFD numerical tank. Corresponding model tests are also implemented to verify and validate against the numerical solutions. The effects of the sloping seafloor, as well as the water depth, on the hydrodynamic coefficients are investigated based on the validated CFD modeling. Both numerical and experimental results indicate that the acquisition of the linear and quadratic damping coefficients is sensitive to the data-processing and identification approaches. For the case studied in present paper, the identification errors introduced by the conventional method are 1.5 % while they are 0.5 % using the direct integration method. The quadratic damping coefficient for heave mode decreases about 10.4 % when the sloping angle increases from 0 to 6 deg.

Prediction of the response of nonlinear dynamical systems under interacting parametric and external excitations is important in designing systems such as sensors, amplifiers or energy harvesters, to achieve the desired performance. This paper concerns the nonlinear forced Mathieu equation, with linear damping and a 2:1 ratio between the parametric and external excitation frequencies. The Method of Varying Amplitudes (MVA) is employed to derive approximate analytical expressions for the response of the system. Both single-term and double-term solutions are developed: it is seen that, employing the double-term approximation, the MVA can accurately predict the response of the system over a wide range of frequencies and system parameters, showing a maximum of 0.2% deviation from numerical results obtained by direct integration of the equation of motion. This is in contrast with most of the available theoretical approaches such as the conventional Method of Multiple Scales, which can predict the response accurately only for a narrow range of system parameters and excitation frequencies. Furthermore, it is seen that the response is bounded, and analytical expressions for the frequency and amplitude of the upper bound are developed: this is unlike other methods which predict unbounded response, unless nonlinear damping is considered. Analytical expressions for the response are developed, and results are verified with numerical results obtained from direct integration of the equation of motion. Numerical examples are presented, showing good agreement with results obtained by the MVA.

Circular and ring plates on a continuous changeable elastic bed are considered. It is assumed that the bed response is described by the Winkler model with a changeable bedding coefficient. An analytical method for axisymmetric bending calculations of plates loaded by a transverse continuously distributed replaceable load is proposed. The method is based on the solution of the corresponding differential bending equation with exchangeable coefficients found by direct integration. The exact solution is expressed through dimensionless fundamental function given in the form of power series of dimensionless parameters with exchangeable coefficients. Recurrence integral relations were obtained to determine the exchangeable coefficients of the series. As a result, formulas for deflections and internal forces, which fully characterize the plate’s stress-strain state, in the case where the given load and bed factor are arbitrary continuous functions, are obtained in analytical form. The method can be applied for any given limit conditions on the plate contours. In fact, calculations for axisymmetric bending of plates are reduced to determination of unknown integration constants from the given limit conditions and numerical realization of the obtained solutions. The derived general formulas are adaped for the practical case with the bedding factor and load given by polynomials. It is shown that, in this case, the dimensionless fundamental functions can be represented by power series. To calculate the coefficients of the power series, the corresponding recurrence relations are derived. Compared with approximate methods, the proposed analytical method allows one to obtain a more realistic bending behavior, being a lucrative alternative to the application of approximate methods when solving this class of problems.

In this paper, the considered two-DOF system consists of a linear oscillator (LO) under external harmonic excitation and an attached lightweight nonlinear energy sink (NES) with local potential and geometrically nonlinear damping. With the application of complex-averaging method, the steady-state dynamical behavior of the system is investigated by the slow invariant manifold, folding singularities and equilibrium points. Different scenarios of strongly modulated responses are presented based on the geometry of SIM, and the numerical simulation results are in consistent with the analytical prediction. The incremental harmonic balance method is applied to detect the frequency response curves of the system around the fundamental resonance, and the accuracy of the theoretical analysis is fully verified by the numerical results obtained by direct integration of equations of motion of the system. It is demonstrated that the increase in external forcing amplitude, global nonlinear stiffness and local nonlinear stiffness can drive the frequency response curves move toward the right and widen the frequency bandwidth of the coexistence of multiple steady-state response regimes, while the increase in nonlinear damping the reverse. The numerical simulation results also show that the addition of geometrically nonlinear damping and local potential in the proposed NES can drastically enhance the capacity of the nonlinear vibration absorber to suppress the shock-induced response of the LO, and the proposed NES is effective for a comparatively broad range of applied impulsive energies, particularly for the high impulsive energies.

For a pulsating free surface source in a three-dimensional finite depth fluid domain, the Green function of the source presented by John [F. John, On the motion of floating bodies II. Simple harmonic motions, Communs. Pure Appl. Math. 3 (1950) 45-101] is superposed as the Rankine source potential, an image source potential and a wave integral in the infinite domain $(0, \infty)$. When the source point together with a field point is on the free surface, John's integral and its gradient are not convergent since the integration $\int^\infty_\kappa$ of the corresponding integrands does not tend to zero in a uniform manner as $\kappa$ tends to $\infty$. Thus evaluation of the Green function is not based on direct integration of the wave integral but is obtained by approximation expansions in earlier investigations. In the present study, five images of the source with respect to the free surface mirror and the water bed mirror in relation to the image method are employed to reformulate the wave integral. Therefore the free surface Green function of the source is decomposed into the Rankine potential, the five image source potentials and a new wave integral, of which the integrand is approximated by a smooth and rapidly decaying function. The gradient of the Green function is further formulated so that the same integration stability with the wave integral is demonstrated. The significance of the present research is that the improved wave integration of the Green function and its gradient becomes convergent. Therefore evaluation of the Green function is obtained through the integration of the integrand in a straightforward manner. The application of the scheme to a floating body or a submerged body motion in regular waves shows that the approximation is sufficiently accurate to compute linear wave loads in practice., Comment: 24 pages, 7 figures

An analytical solution is presented for the 3D static response of variable stiffness non-uniform composite beams. Based on Euler-Bernoulli theory, a set of governing differential equations are obtained, in which four degrees of freedom are fully coupled. For the variable stiffness beam, the governing field equations have variable coefficients reflecting the stiffness variation along the beam. Using the direct integration technique, the general analytical solution is derived in the integral form and the closed-form expressions of the obtained solutions are presented employing a series expansion approximation. The series expansion representation enables the proposed approach to be applicable for variable stiffness composite beams with arbitrary span-wise variation of properties. As an alternative solution, the Chebyshev collocation method is applied to the proposed formulation to verify the results obtained from the analytical solution. A number of variable stiffness composite beams made by fibre steering with various boundary conditions and stacking sequences are considered as the test cases. The static response are presented based on the analytical solution and Chebyshev collocation method and excellent agreement is observed for all test cases. The proposed model presents a reliable and efficient approach for capturing the complicated behaviour of variable stiffness non-uniform composite beams.

The existing non-paraxial expression of audio sounds generated by a parametric array loudspeaker (pal) is hard to calculate due to the fivefold integral in it. A rigorous solution of the Westervelt equation under the quasilinear approximation is developed in this paper for circular PALs by using the spherical harmonics expansion, which simplifies the expression into a series of threefold summations with uncoupled angular and radial components. The angular component is determined by Legendre polynomials and the radial one is an integral involving spherical Bessel functions, which converge rapidly. Compared to the direct integration over the whole space, the spherical expansion is rigorous, exact, and can be calculated efficiently. The simulations show the proposed expression can obtain the same accurate results with a speed of at least 15 times faster than the existing one.

In a previous paper, one of us (UL) constructed an exact analytical expression for the irregular (Jost) solution, off the energy shell, for the Hulthen potential by using the method of ordinary differential equation along with the properties of certain special functions of mathematical physics. In the present text, closed form expressions for the off-shell irregular solution and half-off-shell Tmatrix for motion in the Hulthen potential are re-derived by using the direct integration approach to the problem under consideration. We demonstrate the usefulness of our expressions through some model calculations and achieve excellent agreement between bound-and scattering-state observables and standard results.

In this work, nonlinear free vibration of electrostatically actuated micromechanical system has been investigated. These nonlinear solutions are analytically derived. An undamped spring-mass model and dynamic equation of motion are employed for the parallel-plate microelectrostatic actuator. The first analytical approximate period and periodic solution of the mentioned problems are constructed with the help of Galerkin’s method and coupling of Galerkin’s with Newton’s method. In addition, the second and third analytical approximate solutions are also derived. The “exact” solutions are obtained from direct integration. In this work, the well-known static and dynamic pull-in parameters are discussed. Phase trajectory becomes non-periodic with the dynamic pull-in point. The approximate and “exact” solutions of period, phase trajectory, and time-dependent displacement of microelectrostatic actuators at various physical parameters are obtained and compared. It is found that the analytical approximate solutions are not only an explicit expression but also have excellent precision for a wide range of nonlinear vibration with different amplitudes.

A consistent co-rotational formulation and numerical procedure for the dynamic analysis of planar Timoshenko beam is presented in detail to demonstrate the finite element analysis techniques of co-rotational formulation for nonlinear dynamic analysis of beam structures. The inertia nodal forces and deformation nodal forces are systematically derived by consistent linearization of the fully geometrically nonlinear beam theory by using the d’Alembert principle and the virtual work principle. The nodal coordinates, incremental displacements and rotations, velocities, accelerations and the equations of motion of the system are defined in terms of a fixed global coordinate system, while the total strains in the beam element are measured in element coordinates which are constructed at the current configuration of the beam element. A practical numerical procedure is proposed to determine the current end cross-section coordinates, element coordinates, and element deformations corresponding to an incremental displacement. An incremental-iterative method based on the Newmark direct integration method and the Newton-Raphson method is employed here for the solution of the 284nonlinear dynamic equilibrium equations. Numerical examples are presented to demonstrate the effectiveness of the proposed method and investigate the effect of the shear deformafion on the dynamic response of the beam structures.

This paper evaluates nonlinear free vibrations of Levy plates using Weak-Form variational principle in algebraic polynomial displacement functions. The energy functional of the plate problem was formulated using Weak-Form variational technique on the integral function of the Von Karman thin plate differential equations. The displacement functions were developed based on static deflection configurations of orthogonal beam network. The process of repeated direct integration on compatibility equation was used to determine the algebraic expressions for stress function. The amplitude of deflection which directly influences the geometric nonlinearity of the plate was determined using integration process on energy functional based on static equilibrium equations. The modal combination method was used to develop the stiffness and mass matrices respectively from the expressions of energy functional based on dynamic equilibrium equations. The numerical values of amplitudes of deflection at various aspect ratios were computed. Also, the first four nonlinear natural frequencies at various aspect ratios were numerically computed. The validation of the present study’s results using the results from previous work found in literature shows satisfactory convergence, with an absolute mean error of 0.186 %. Conclusively, the application of Weak-Form variational principle in polynomial displacement functions provides satisfactory approximation to nonlinear dynamic analysis of Levy plates.

In this paper, a novel version of orthonormal polynomial series expansion method is developed to analyze vibrations of bottom hull panels due to moving impacting fluid loads. The initial phase of slamming is considered in the presented mathematical model, wherein, an elastic linear Euler-Bernoulli beam of uniform thickness mimics the bottom panel structural behaviour. The framework developed, suppresses the Gibbs phenomenon, and captures the beam deflection, flexural moment, and shear force by direct integration. The proposed solution exhibits a higher convergence rate, compared to the conventional method, in terms of the number of involved shape functions.

We developed a technique for the construction of solutions of axisymmetric problems of elasticity and thermoelasticity in stresses for a space and a half space whose elastic properties are arbitrary functions of the coordinate z. By using the direct integration method and the Hankel integral transformation, the problems are reduced to governing integral equations accompanied by a local boundary condition and an integral condition in the case of the half space. The solutions of the deduced equations are constructed in the explicit form by using the resolvent-kernel method.

A static fourth-order Euler-Bernoulli beam equation, corresponding to a negative fractional power law for the applied load, has been completely integrated in this paper. For this equation the Lie symmetry and the Noether symmetry algebras are isomorphic to sl ( 2 , R ) . Due to this algebra is nonsolvable, the symmetry reductions that have been employed so far in the literature fail to obtain the complete solution of the equation. A new strategy to obtain a third-order reduction has been performed, which provides, by direct integration, one of the first integrals of the equation. This first integral leads to a one-parameter family of third-order equations which preserves sl ( 2 , R ) as symmetry algebra. From these equations, three remaining functionally independent first integrals have been computed in terms of solutions to a linear second-order equation and, as a consequence, the exact general solution has been obtained. As far as we know, this has not been previously reported in the literature. The general solution can be expressed in parametric form in terms of a fundamental set of solutions to a one-parameter family of Schrodinger-type equations.

This paper addresses the dynamics of locally-resonant sandwich beams, where multi-degree-of-freedom viscously-damped resonators are periodically distributed within the core matrix. On adopting an established model in the literature, which consists of an equivalent single-layer Timoshenko beam coupled with mass-spring-dashpot subsystems representing the resonators, novel and exact analytical expressions are presented for the frequency response and modal response under arbitrary loads. Specifically, the frequency response is built by a direct integration method, while the modal impulse and frequency response functions are derived by a complex modal analysis approach, upon introducing orthogonality conditions pertinent to the complex modes. The expressions obtained for frequency response and modal response hold for any number of resonators and degrees of freedom within the resonators, any number of loads and positions of the loads relative to the resonators. In the proposed complex modal analysis approach, the challenging issue of calculating all complex eigenvalues, without missing anyone, is solved applying a recently-introduced contour-integral algorithm to an exact dynamic stiffness matrix that, here, is built with size depending only on the number of degrees of freedom at the beam ends, regardless of the number of resonators and degrees of freedom within the resonators. Numerical applications prove exactness and robustness of the proposed framework.

In this article, we examine a (3+1)-dimensional generalized breaking soliton equation which is highly applicable in the fields of engineering and nonlinear sciences. Closed-form solutions in the form of Jacobi elliptic functions of the underlying equation are derived by the method of Lie symmetry reductions together with direct integration. Moreover, the (G&prime, /G)-expansion technique is engaged, which consequently guarantees closed-form solutions of the equation structured in the form of trigonometric and hyperbolic functions. In addition, we secure a power series analytical solution of the underlying equation. Finally, we construct local conserved vectors of the aforementioned equation by employing two approaches: the general multiplier method and Ibragimov&rsquo, s theorem.

In this article, we consider the model of electrical transmission line for exact solutions and conservations laws. For this, lie point symmetries are obtained. It is noticed that the equation under study is nonlinearly self-adjoint. This property is useful to calculate more conservation laws. Conservation laws are computed by using the new general conservation theorem of Ibragimov and multiplier approach. Some new exact solutions are achieved using direct integration and $$ \left( {1/G^{\prime}} \right) $$ —expansion methods.

An efficient technique for thermoelastic analysis of inhomogeneous anisotropic solids is suggested within the framework of three-dimensional formulation. By making use of the direct integration method, a system of governing equations is derived in order to solve three-dimensional problems of elasticity and thermoelasticity for transversely isotropic inhomogeneous solids with elastic and thermo-physical properties represented by differentiable functions of the variable in the direction that is transversal to the plane of isotropy. By implementing the relevant separation of variables, the obtained equations can be uncoupled and reduced to second-kind integral equations for individual stress-tensor components and the total stress, which represents the trace of the stress tensor. The latter equations can be attempted by any of the numerical, analyticalnumerical, or analytical means available for the solution of the second-kind integral equations. In order to construct the solutions in an explicit form, an advanced solution technique can be developed on the basis of the resolvent-kernel method implying the series representation by the recurring kernels, computed iteratively by the original kernel of an integral equation.

In this paper, the periodic solution for the local fractional Boussinesq equation can be obtained in the sense of the local fractional derivative. It’s given by applying direct integration with symmetry condition. Meanwhile, the periodic solution of the non-differentiable type with generalized functions defined on Cantor sets is analyzed. As a result, we have a new point to look the local fractional Boussinesq equation through the local fractional derivative theory.

Previous work on scattering by Bessel beams shows that expansion of incident sound fields in term of these beams has application to scattering [P. L. Marston, J. Acoust. Soc. Am. 122, 247–252 (2007)]. In this work, an expression for the expansion coefficients of propagating, axisymmetric, sound fields are derived. In this paper, this expression is applied to a linear focused axisymmetric sound field and is expanded in terms of Bessel beam components. This is done for focused beams radiated from a spherical cap source. A physical optics model is applied to sound propagation close to the source to facilitate the calculation of the Bessel beam expansion coefficients. This type of model is useful for focused scattering [P. L. Marston and D. S. Langley, J. Acoust. Soc. Am. 73, 1464–1475 (1983)]. Once the expansion coefficients are found, the sound field can be evaluated by superposition. The model agrees approximately with Chen, Schwarz, and Parker [J. Acoust. Soc. Am. 94, 2979–2991 (1993)] and O'Neil [J. Acoust. Soc. Am. 21, 516–526 (1949)] on axis and with direct integration of a Kirchhoff integral both on and off axis. This type of expansion will have applications to scattering problems.Previous work on scattering by Bessel beams shows that expansion of incident sound fields in term of these beams has application to scattering [P. L. Marston, J. Acoust. Soc. Am. 122, 247–252 (2007)]. In this work, an expression for the expansion coefficients of propagating, axisymmetric, sound fields are derived. In this paper, this expression is applied to a linear focused axisymmetric sound field and is expanded in terms of Bessel beam components. This is done for focused beams radiated from a spherical cap source. A physical optics model is applied to sound propagation close to the source to facilitate the calculation of the Bessel beam expansion coefficients. This type of model is useful for focused scattering [P. L. Marston and D. S. Langley, J. Acoust. Soc. Am. 73, 1464–1475 (1983)]. Once the expansion coefficients are found, the sound field can be evaluated by superposition. The model agrees approximately with Chen, Schwarz, and Parker [J. Acoust. Soc. Am. 94, 2979–2991 (1993)] and O'Neil [J. Acou...

Explicit expressions for Eshelby’s tensor of an elliptic inclusion in orthotropic materials are derived in this paper. The main approach is based on the general solution for elliptic-hole problems in orthotropic materials. The derivation based on a direct integration of elastic Green’s function along the elliptic contour is also used to illustrate a certain useful connection between these two approaches; this connection has not been fully recognized and published before. The theoretical results are checked with those obtained via the Fourier transform method and those found by direct numerical integration.

The present paper is concerned with the numerical modelization of the transient dynamic response of a simply supported Euler–Bernoulli elastic beam resting on a Winkler-type foundation, under the action of a transverse concentrated load, moving at a constant velocity along the beam, displaying an harmonic-varying magnitude in time. The elastic foundation, assumed as homogeneous in space, behaves according to a bilinear constitutive law, characterized by two different stiffness coefficients in compression and in tension. A finite element method approach coupled with a direct integration algorithm is developed for efficiently tracing the nonlinear dynamic response of the beam-foundation system. An original automated procedure is set, as being apt to resolve all required space/time discretization issues. Extensive parametric numerical analyses are performed to investigate how the frequency of the harmonic moving load amplitude and the ratio between the foundation’s moduli in compression and in tension affect the so-called critical velocities of the moving load, leading to high transverse beam deflections. Analytical interpolating expressions are proposed and fitted for the achieved two-branch critical velocity trends. The present outcomes shall reveal potential practical implications in scenarios of contemporary railway engineering, especially in terms of lowering down the admissible high-speed train velocities, as for structural requirement or preventing potential passenger discomfort.

An accurate analytical solution of the stationary problem of heat conductivity for a multilayer structure with heat sources under asymmetric boundary conditions of the third kind is obtained based on the use of the theory of generalized functions. Using the asymmetric unit function (Heaviside function), the problem for a multilayer structure is reduced to a single-layer with discontinuous (piecewise-continuous) medium properties. By introducing a new independent variable, the nonlinear differential equation (nonlinearity of the second kind—the thermal-physical properties depend on the spatial variable) is reduced to a linear one, direct integration of which allows obtaining an accurate analytical solution of the problem considered. Examples of investigations of the temperature fields for some certain multilayer structures are given.

This paper analyses the performance of the main radial basis functions in the formulation of the Boundary Element Method (DIBEM). This is an alternative for solving problems modeled by non-adjoint differential operators, since it transforms domain integrals in boundary integrals using radial basis functions. The solution of eigenvalue problem was chosen to performance evaluation. Natural frequencies are calculated numerically using several radial functions and their accuracy is evaluated by comparison with the available analytical solutions and with the Finite Element Method as well. The standard radial basis functions have presented similar performance to compact radial functions, being even slightly superior.

In this paper we present a new class of direct numerical integrators of hybrid type for special third order ordinary differential equations (ODEs), y ′ ′ ′ = f ( x , y ) ; namely, hybrid methods for solving third order ODEs directly (HMTD). Using the theory of B-series, order of convergence of the HMTD methods is investigated. The main result of the paper is a theorem that generates algebraic order conditions of the methods that are analogous to those of two-step hybrid method. A three-stage explicit HMTD is constructed. Results from numerical experiment suggest the superiority of the new method over several existing methods considered in the paper.

The non-linear partial differential (2+1) dimensional Breaking Soliton equation is studiedby using the direct integration and homotopy perturbation method. In this study, we use directintegration to obtain the known solution in the literature in practical and shortest way by assigningsome special values to the constants in the solutions of the (2+1) dimensional Breaking Solitonequation. We also obtain same type solution for (2+1) dimensional Breaking Soliton equation byusing the homotopy perturbation method with one iteration. Similarly, same type solutions can bedone different methods such as (G'/G)-expansion method.

Two variants of a refined theory for calculation of the rectangular orthotropic plates stress-strain state are represented. The plate's state equations are presented in the form three-dimensional equations of elasticity theory. The components of the plate's stress-strain state are received as the polynomial func- tions on the coordinate which is normal to the middle plane. These functions are one or two degree higher than in the Kirchhoff-Love theory are used. The virtual displacements principle is applied to obtain the two-dimensional equations and its natural boundary conditions. The modified boundary conditions for standard cases of the plate mounting are formulated. Calculation of plate stress-strain is carried out by using Laplace transform, and then the number of arbitrary constants in the integration of differential equations systems thus is twice reduced. One of the refined theory distinctive features consist in direct integration of the three dimensional elasticity problems equilibrium equations at transverse normal and tangential stresses determination. As an example, the paper considers the calculation of a rectangular isotropic plate's stress-strain under a local load. The results obtained by the refined theories and by the classical theory are compared. The essential contribution of normal transverse stress of type "boundary layer" to the general stress- strain state of a plate is shown. The received results can be used in calculations and at tests for strength and durability of aviation and space-rocket and also engineering structures of different destination

Analytical solutions to the axisymmetric heat-conduction and thermoelasticity problems for a long solid cylinder are constructed for arbitrary variation of the thermomechanical properties within the radial coordinate. By making use of the direct integration method, the steady-state heat-conduction equation is reduced to an integral equation of second kind. The numerical, analytical-numerical and analytical methods for solving the latter equation are suggested. The one-to-one relations between the stresses and displacements on the boundary are established, which allows for the reduction of the thermoelasticity problems with various types of the boundary conditions to the one in terms of stresses. The explicit expressions for the stress-tensor and displacement-vector components are obtained and analyzed for different types of inhomogeneity.

In this study, the weak-form finite element model for bending of beams considering constrained micro-rotation and rotation gradient-dependent potential energy is developed for the moderate rotationcase. The governing equations for a general higher-order beam theory with the von Karman geometric nonlinearity are derived from the principle of virtual displacements. The formulated finite element model is valid for homogeneous, orthotropic, and functionally graded classical and microstructure-dependent beams. Further, the specialization of the theory to various existing beam theories is also presented. The analytical solution for the simply supported beam for the linear case is also derived. In the numerical examples presented, the stiffening effect due to the consideration of microstructure in the micro-beam is illustrated. The parametric effect of the material length scale on the bending moment and stress is also investigated.

In this work, stability of thin flexible Bernoulli-Euler beams is investigated taking into account the geometric non-linearity as well as a type and intensity of the temperature field. The applied temperature field T(x,z) is yielded by a solution to the 2D Laplace equation solved for five kinds of thermal boundary conditions, and there are no restrictions put on the temperature distribution along the beam thickness. Action of the temperature field on the beam dynamics is studied with the help of the Duhamel theory, whereas the motion of the beam subjected to the thermal load is yielded employing the variational principles. The heat transfer (Laplace equation) is solved with the use of the finite difference method (FDM) of the third-order accuracy, while the integrals along the beam thickness defining the thermal stress and moments are computed using Simpson's method. Partial differential equations governing the beam motion are reduced to the Cauchy problem by means of application of FDM of the second-order accuracy. The obtained ordinary differential equations are solved with the use of the fourth-order Runge-Kutta method. The problem of numerical results convergence versus a number of beam partitions is investigated. A static solution for a flexible Bernoulli-Euler beam is obtained using the dynamic approach based on employment of the relaxation/set-up method. Novel stability loss phenomena of a beam under the thermal field are reported for different beam geometric parameters, boundary conditions, and the temperature intensity. In particular, it has been shown that stability of the flexible beam during heating the beam surface essentially depends on the beam thickness.

A family of k -step block multistep methods where the main formulas are of Falkner type is proposed for the direct integration of the general second order initial-value problem where the differential equation is of the general form y ź = f ( x , y , y ' ) . The two main Falkner formulas and the additional ones to complete the block procedure are obtained from a continuous approximation derived via interpolation and collocation at k + 1 points. The main characteristics of the methods are discussed through their formulation in vector form. Each method is formulated as a group of 2 k simultaneous formulas over k non-overlapping intervals. In this way, the method produces the approximation of the solution simultaneously at k points on these intervals. As in other block methods, there is no need of other procedures to provide starting approximations, and thus the methods are self-starting (sharing this advantage of Runge-Kutta methods). The resulting family is efficient and competitive compared with other existing methods in the literature, as may be seen from the numerical results.

The paper focuses on the application of the sampling surfaces (SaS) method to exact solutions of elasticity for laminated plates. The SaS method is based on choosing the arbitrary number of SaS parallel to the middle surface in order to introduce the displacements of these surfaces as basic plate unknowns. Such choice of unknowns with the use of the Lagrange polynomials in assumed approximations of displacements through the layer thicknesses leads to a compact laminated plate formulation. The feature of the proposed approach is that all SaS are located inside the layers at Chebyshev polynomial nodes. The use of outer surfaces and interfaces is avoided that makes possible to minimize uniformly the error due to the Lagrange interpolation. Therefore, the strong SaS formulation based on direct integration of the equilibrium equations of elasticity can be applied efficiently to the obtaining of 3D exact solutions for laminated plates.

The meshless analog equation method, a purely meshless method, is applied to the buckling analysis of moderately thick plates described by Mindlin’s theory and resting on two-parameter elastic foundation (Pasternak type). The method is based on the concept of the analog equation, which converts the three governing second-order partial differential equations (PDEs) in terms of the three plate displacements (transverse displacement and two rotations) into three substitute equations, the analog equations. The analog equations constitute a set of three uncoupled Poisson’s equations under fictitious sources, which are approximated by multi-quadric radial basis functions (MQ-RBFs) series. This enables the direct integration of the analog equations and allows the representation of the sought solution by new RBFs series. These RBFs approximate accurately not only the displacements but also their derivatives involved in the governing equations. Then, inserting the approximate solution in the original PDEs and in the associated boundary conditions (BCs) and collocating at mesh-free nodal points, a generalized eigenvalue problem is obtained, which allows the evaluation of the buckling load and the buckling modes. The studied examples demonstrate the efficiency of the presented method that is its ability to solve accurately and in a straightforward way difficult engineering problems.

In this paper, steady bifurcation and solitary waves of the perturbed PHI-four equation are investigated. Steady bifurcation, bistability and bi-instability are analyzed by selecting the perturbation term as the bifurcation control parameter. Furthermore, we obtain different conditions for the existence of solitary or shock waves and all kinds of the solitary and shock wave solutions by direct integration.

Seeking to validate a solution technique for Helmholtz problems, the Boundary Element Method with Direct Integration, which applies radial basis functions to approach the inertia term, is used to solve numerically problems governed by Helmholtz Equation. The standard Boundary Element formulation that employs the fundamental solution correlated to the Helmholtz Equation and has the frequency of excitation as argument is used for comparison. Thus, examples comprising the direct solution of Helmholtz problems are solved with both Boundary Element formulations and then their results are compared with available analytical solutions.

We obtain the exact, closed-form solutions for the electromagnetic field around a finite-length transmitter in a conducting homogeneous whole space. The field is obtained for the two cases following after the transmitter is turned on or off. The electric field solution is obtained by direct integration of the source currents along with the Green’s function of the diffusion equation, and the magnetic field is obtained by the direct integration of Ampere’s law.

Present investigation is concerned with the free vibration property of a beam with periodically variable cross-sections. For the special geometry characteristic, the beam was modelled as the combination of long equal-length uniform Euler-Bernoulli beam segments and short equal-length uniform Timoshenko beam segments alternately. By using continuity conditions, the hybrid beam unit (ETE-B) consisting of Euler-Bernoulli beam, Timoshenko beam and Euler-Bernoulli beam in sequence was developed. Classical boundary conditions of pinned-pinned, clamped-clamped and clamped-free were considered to obtain the natural frequencies. Numerical examples of the equal-length composite beam with 1, 2 and 3 ETE-B units were presented and compared with the equal-length and equal-cross-section Euler-Bernoulli beam, respectively. The work demonstrates that natural frequencies of the composite beam are larger than those of the Euler-Bernoulli beam, which in practice, is the interpretation that the inner-welded plate can strengthen a hollow beam. In this work, comparisons with the finite element calculation were presented to validate the ETE-B model.

Expressions for the stored energy and Q factor of an antenna in terms of its radiated spherical modes and reflection coefficient have been extended to include the effects of loss. Three variations are given for excitation by matched, voltage, or current generators, and the importance of the proper phase reference for the radiated fields is emphasised. Two low-Q, self-resonant spherical antennas are used to illustrate the application of the energy formulae with and without loss. The results for these examples are validated by direct integration of the electric energy and by self-consistency of the radiation and dissipation terms. Approximation equations appropriate for processing experimental data are derived, and their predictions are compared with the actual Q’s for the examples. Estimates of the lossless Q’s are determined from the lossy data.

The objective of this paper is to present a finite element solution for the wave propagation problems with a reduction of the velocity dispersion and spurious reflection. To this end, a high-order two-step direct integration algorithm for the wave equation is adopted. The suggested algorithm is formulated in terms of two Hermitian finite difference operators with a sixth-order local truncation error in time. The two-node linear finite element presenting the fourth-order of local truncation error is considered. The numerical results reveal that although the algorithm competes with higher-order algorithms presented in the literature, the computational effort required is similar to the effort required by the average acceleration Newmark method. More than that, the integration with the lumped mass model shows similar results to the integration using the average acceleration Newmark for the consistent mass model, which involves a higher number of computational operations.

The paper presents the mathematical and numerical formulation of recently articulated finite element method for flexible structures. The Absolute Nodal Coordinate Formulation (ANCF), proposed by A. Shabana helps to numerically articulate the large deformations, reportedly with precision. Renowned finite element abstraction based on Euler Bernoulli's beam element is used to drive the ANCF approach of beam element. Finite coordinates and slopes are used as nodal variable with respect to global frame of reference. Each node is completely defined by four nodal variables in a planar 2-dimensional formulation. The proposed formulation facilitates to extract the rigid body modes with accuracy. Analytical formulation of Absolute Nodal Coordinate Formulation (ANCF) uses constant mass matrix and generalized gravity but the strain matrix is highly nonlinear as it originates from green Langrage strain. Legendre's quadration integration is used to extract the elastic forces. Dynamic system is solved through direct integration technique by simultaneously solving equations of motions. The joints between multibody are modelled by kinematic constraint as a consequence, nonlinear algebraic equations are formed. Ordinary differential equation (ODE) and (Algebraic Equations) AE are simulated by Newmark's implicit integration. The paper presents the results from 2-dimensional ANCF applications like cantilever beam, pendulum and four bar mechanism. The paper also presents planar 2-dimensional case of Euler Bernoulli's finite element formulation for comparison and static analysis of cantilever beam. The dynamic analysis is provided for cantilever beam and pendulum and elaborated with the help of time history diagram.

The sign problem of relativistic field theories at finite fermion chemical potential has been approached by deforming the domain of integration into complex field space. We present a method for selecting a deformed manifold of integration which is a local maximum of the average phase, and demonstrate this method on the three-dimensional Thirring model. Finally, we compare the performance of this method, in the heavy-dense limit, to direct integration on the Lefschetz thimbles., Comment: 7 pages; proceedings from LATTICE2018

This paper presents the three-dimensional analysis of laminated composite rectangular plates with general boundary and loading conditions using the strong sampling surfaces (SaS) formulation and the extended differential quadrature (EDQ) method proposed by the first author. The strong SaS formulation is based on the choice of SaS parallel to the middle surface and located at Chebyshev polynomial nodes to introduce the displacements of these surfaces as plate unknowns. This choice of unknowns with the use of Lagrange polynomials in the through-thickness approximations of displacements, strains and stresses leads to an efficient laminated plate formulation. The outer surfaces and interfaces are not included into a set of SaS that makes it possible to minimize uniformly the error due to Lagrange interpolation. Therefore, the strong SaS formulation based on direct integration of the equilibrium equations of elasticity in the transverse direction in conjunction with the EDQ method can be applied effectively to high-precision calculations for the composite rectangular plates with different boundary conditions. This due to the fact that in the SaS/EDQ formulation the displacements, strains and stresses of SaS are interpolated in a rectangular domain using the Chebyshev-Gauss-Lobatto grid and Lagrange polynomials are also utilized as basis functions. Such a technique allows the use of only first order derivatives in the equilibrium equations that simplifies the implementation of the EDQ method.

The fully anisotropic response of composite beams is an important consideration in diverse applications including aeroelastic responses of helicopter rotor and wind turbine blades. Our goal is to present exact analytical solutions for the first time for coupled deflection of Euler–Bernoulli composite beams. Towards this goal, two approaches are proposed: (1) obtaining the exact analytical solutions directly from the governing equations of Euler–Bernoulli composite beams and (2) extraction of the solutions from Timoshenko composite beam solutions. For the direct solution approach, based on Euler–Bernoulli theory, new variationally-consistent field equations are obtained, in which four degrees of freedom, i.e. in-plane bending, out-of-plane bending, twist and axial elongation are fully coupled. By expressing the coupled system of differential equations in a compact matrix form, a novel expression for the eccentricity of neutral axes from the midplane, as well as the shift in shear centre from the centre of beam, is obtained. This eccentricity matrix serves to decouple the bending in the two principal directions from in-plane and twist deformations. Then, the general closed-form analytical solutions for the decoupled system are derived simply using direct integration. Additionally, the analogous closed-form analytical solutions are retrieved from the previously obtained Timoshenko composite beam solution and it is proven that they are identical to those obtained from the current direct approach for conditions where Euler–Bernoulli beam theory apply. To study the effects of anisotropy, numerical results are obtained for a number of examples with different composite stacking sequences showing various coupled behaviours. The results are compared against the Chebyshev collocation method as well as against less comprehensive analytical solutions available in the literature, noting that excellent agreement is observed, where expected. The present exact solutions can serve as benchmark problems for assessing the accuracy and convergence of various analytical and numerical methods.

This paper studies the Sasa-Satsuma equation, which is used to describe the nonlinear phenomenon of pulse propagation in optical fiber communication. By using the direct integration method, all envelope traveling wave patterns are given, including singular solutions, solitary wave solutions and elliptic function double periodic solutions. These patterns show completely the behaviors of dynamics of the model.

In this article a variable order variable step size technique in backwards difference form is used to solve nonlinear Riccati differential equations directly. The method proposed requires calculating the integration coefficients only once at the beginning, in contrast to current divided difference methods which calculate integration coefficients at every step change. Numerical results will show that the variable order variable step size technique reduces computational cost in terms of total steps without effecting accuracy.

The prototype of mold oscillator that uses a non-sinusoidal generator was modified using Inverse dynamics and Multi-body dynamics. We maintained a basic structure of the mold oscillator including the motor, the gear, the leaf-spring and the caster. To reduce the coupling effect, the non-sinusoidal generator was replaced by a hollow cylinder that had non-sinusoidal surface. The system parameters were obtained from previous research [1], and a new design of the connecting rod between the cylinder and the leaf spring was analyzed by sensitivity analysis. A definition of the detailed system component including gear mesh, and non-sinusoidal surface, Lagrange's equation, Fourier series, and Cam profile synthesis was used to define an equation of motion. The response of oscillating system was derived by a direct integration method. The model was simulated according a specific assumption for each component, and a control system corresponding to real operation was applied. Dynamic responses are displacement, velocity and acceleration of each component. All responses from these simulations were analyzed in the time-frequency region by Fast Fourier transform. Using the model, sensitivity analysis about design variables was also executed about operating performance.

The vibration analysis of bar element systems with continuous variable parameters using the direct integration method is proposed. The method is based on analytical solutions of the appropriate differential equations. By using this method, the problem of free flexural vibrations of a straight bar with arbitrary, continuously varying transverse stiffness and mass per unit length is solved. Formulas in analytical form are derived for dynamic parameters of bar structures, and the procedure for their numerical implementation is presented. Free vibrations of a truncated cone and chimney with a variable wall thickness are studied.

Despite the extensive study of the transient heat conduction analysis for isotropic media by the boundary element method (BEM), its relevant researches in transient heat conduction in anisotropic media still remain relatively scarce indeed. In the time-stepping BEM scheme for the transient heat conduction, the transient effect reveals itself as an additional volume integral that conventionally requires domain discretization for direct integration. However, such domain discretization will destroy the BEM׳s most distinctive nature that only the boundary needs to be discretized. In this paper, the domain integral is analytically transformed to the boundary so that the BEM׳s nature of boundary discretization is completely restored. Moreover, the domain mapping technique is combined with this transformed time-stepping scheme to treat the 2D transient heat conduction in anisotropic media. Without any internal treatments or special approximations like in other schemes, the transformed time-stepping scheme can be applied to effectively solve the problem of transient heat conduction in anisotropic media.