Fourier Series Solution Research Articles

Determining the constant diffusion coefficient by Fourier series solution of the diffusion equation

The purpose of this study was to obtain the diffusion coefficient by calculating directly from the Fourier series solution of the diffusion equation, which has not been studied so far. The Fourier series solution is the sum of a steady-state solution and a transient solution. The steady-state solution is a linear function of distance. The transient solution is the product of a distance function and a time function. The Fourier sine series solution, a function of distance only, is negative for the steady-state solution. For the sublimation diffusion of disperse dye in paste into polyethylene terephthalate (PET) film using the film-roll method, the constant surface concentration and the diffusion distance were determined from the quadratic regression curve for the concentration distribution at a long time. The concentration distribution of the transient solution is a downwardly convex curve with negative values. The concentration distribution of the Fourier series solution is also a downwardly convex curve. The diffusion coefficient in the exponential term, a function of time only, was determined from the value of the exponential term obtained by the surface concentration and the diffusion distance. The constant diffusion coefficients determined from the Fourier series solution are reliable because they are very similar to those obtained from the error function solution and have excellent linearity of the linear regression lines for their Arrhenius plots.

Comparison of formulas derived from Fourier series solution of diffusion equation

The first Fourier series solution satisfying the boundary condition of zero concentration at the diffusion distance was derived by assuming that the solution consists of a steady-state solution and a transient solution. The first total amount which has passed through the first layer surface was derived from the first Fourier series solution. When the diffusion distance is large, the exponential term in the first total amount cannot be zero. For the sublimation diffusion of disperse dye in paste into polyethylene terephthalate film using the film-roll method, the plot of the first total amount against time was in good agreement with the quadratic regression curve. The second Fourier series solution satisfying the boundary condition of constant concentration at the diffusion distance was derived by assuming that the diffusion distance is the thickness of the first layer. The second total amount which has passed through the second layer surface was derived from the second Fourier series solution and is a linear function of time because the exponential term is zero due to the very small diffusion distance, the thickness of the first layer. The plot of the second total amount against time was in good agreement with the linear regression line. The time-lag formula was derived from the second total amount. The diffusion coefficients determined from the second total amount are much larger than those of the other three types. All four types showed excellent linearity in Arrhenius plot, but the activation energy for the time-lag formula is somewhat greater than the rest.

Investigation of moving load-induced vibrations in layered functionally graded Terfenol-D beams: a differential quadrature method and analytical approach

The paper investigates the potential of the full-sinusoidal Fourier series as a solution form for the deflection of layered functionally graded beams associated with smart actuators, drawing on the fundamental principles of classical Fourier series theory. The current method simplifies the difficult beam problem to a set of linear algebraic equations by using the Duhamel integration technique and the orthogonality of the trial function. The study takes into account generalized boundary conditions and moving forces, which are seldom discussed in previous research. Under the action of a moving load, the boundary value problem for a functionally graded beam integrated with Terfenol-D is efficiently addressed using the approach of combined differential and integral quadrature. The generalized boundary conditions may be easily achieved by adjusting the stiffness of the restraining springs. The significant agreement between the differential quadrature solution and the Fourier series solution underlines the efficiency and accuracy of both methods. Furthermore, the influences of various crucial physical characteristics on the natural frequencies and the essential flow velocities are explored, including boundary stiffness, foundation parameters, and geometric parameters.

Effects of pore-fluid pressure on the motion of debris flows

ABSTRACT Pore-fluid pressure mediates the shear strength between soil particles and significantly controls the movement of debris flows. This study analyzes the effects by proposing a numerical model to solve the pore-fluid diffusion equation coupled to changes in surge motion of debris flows. The model was established by combining a Lagrangian method solving the depth-averaged one-dimensional equations for the debris-flow motion and a Fourier-series solution for the pore-fluid diffusion equation. Owing to the source term related to the change in surge depth, the Fourier-series solution shows that the excess pore-fluid pressure near the bed dissipates faster and exhibits a non-monotonic profile. The results of numerical calculations reveal that the diffusivity of excess pore-fluid pressure and the liquefaction condition have significant impacts on the depth profiles and velocities of debris-flow surges.

An Analytical Solution for the Bending of Anisotropic Rectangular Thin Plates with Elastic Rotation Supports

The mechanical analysis of thin-plate structures is a major challenge in the field of structural engineering, especially when they have nonclassical boundary conditions, such as those encountered in cement concrete road slabs connected by transfer bars. Conventional analytical solutions are usually limited to classical boundary conditions—clamped support, simple support, and free edges—and cannot adequately describe many engineering scenarios. In this study, an analytical solution to the bending problem of an anisotropic thin plate subjected to a pair of edges with free opposing elastic rotational constraints is found using a two-dimensional augmented Fourier series solution method. In the derivation process, the thin-plate problem can be transformed into a problem of solving a system of linear algebraic equations by applying Stoke’s transform method, which greatly reduces the mathematical difficulty of solving the problem. Complex boundary conditions can be optimally handled without the need for large computational resources. The paper addresses the exact analytical solutions for bending problems with multiple combinations of boundary conditions, such as contralateral free–contralateral simple support (SFSF), contralateral free–contralateral solid support–simple support (CFSF), and contralateral free–contralateral clamped support (CFCF). These solutions are realized by employing the Stoke transformation and adjusting the spring parameters in the analyzed solutions. The results of this method are also compared with the finite element method and analytical solutions from the literature, and good agreement is obtained, demonstrating the effectiveness of the method. The significance of the study findings lies in the simplification of complex nonclassical boundary condition problems using a simple and reliable analytical method applicable to a wide range of engineering thin-plate structures.

An improved Fourier series solution for vibration analysis of elastically restrained beams in Rayleigh–Ritz method

A combination of Fourier cosines series and polynomials (4 terms, 4th order) in Rayleigh–Ritz method is proposed to solve the vibration problem of a generally restrained beam. The characteristics of auxiliary polynomials, which actually are determined by homogeneous boundary conditions of the beam under consideration, can directly change the convergence properties and numerical instabilities of vibration results significantly. Fast convergence and reliability of the proposed polynomials are illustrated with numerical examples in comparisons with available results.

Nonlinear stochastic wave equations in 1D with fractional Laplacian, power-law nonlinearity and additive [formula omitted]-regular noise

A qualitative study of nonlinear, 1D stochastic fractional wave equations with dissipative nonlinearities of power-law form is conducted on (t,x)∈[0,+∞)×Dutt+σ2(−uxx)α−a1u+a2‖u‖L2(D)ρu−κut=b0∂W0∂t on (t,x)∈[0,+∞)×D with D=[0,L], where positive fractional α-powers of Laplace operator are allowed, perturbed by additive space–time random noise W0 with fairly general covariance operator Q with finite trace(Q)<+∞ (i.e. Q-regularity). The Q-regular space–time noise W0, which is white in time and generally spatially correlated, is supposed to have a Fourier expansion along the eigenfunctions of the Laplace operator under nonrandom, Dirichlet- and Neumann-type homogeneous boundary conditions. We focus on studying the generalized energy functional V incorporating the fractional diffusive energy part and the energy part which is due to the viscous damping force both in continuous and discrete time. In continuous time, the technique of Fourier expansions and appropriate truncation by finite-dimensional systems leads to the control on the generalized energy functional, and hence we may verify existence, uniqueness and boundedness of moments of Fourier series solutions u. The total mean energy E[E(t)] cannot grow more than linearly in time t (with or without damping). Moreover, under the absence of damping (i.e. κ=0), E[E(t)] is governed by a kind of trace formula in case of additive (state-independent), Q-regular space–time noise W0. For numerical computations and more adequate discretization, we suggest to take nonstandard, partial-implicit midpoint-type methods for the Fourier coefficients. These semi-analytical numerical methods (approximate Fourier series) possess the property of conserving the expected, total energy with random initial data under the absence of nonlinearities. Eventually, we estimate probabilities of large fluctuations of some interesting functionals.

Laplace Transform Methods for Transient Diffusion; or, Some Good Questions from Ralph White

Motivated by years of correspondence with Prof. Ralph White, I discuss two unconventional ways to solve diffusion problems with Laplace transforms. A method to derive error-function series, alternatives to Fourier series that converge rapidly and avoid the Gibbs phenomenon at short times, is illustrated by example. It is shown how Mittag-Leffler partial-fractions expansions can facilitate derivations of Fourier-series solutions from the same starting point. Several basic problems pertinent to electrochemical transport are analyzed, culminating in the development of a modified Cottrell equation applicable to thin films of unsupported electrolytic solutions sandwiched between planar electrodes.

The evolution Navier–Stokes equations in a cube under Navier boundary conditions: rarefaction and uniqueness of global solutions

We study the evolution Navier–Stokes equations in a cube under Navier boundary conditions. For the related stationary Stokes problem, we determine explicitly all the eigenvectors, eigenvalues and the corresponding Weyl asymptotic. We introduce the notion of rarefaction, namely families of eigenvectors that weakly interact with each other through the nonlinearity. By combining the spectral analysis with rarefaction, we expand the solutions in Fourier series, making explicit some of their properties. We then suggest several new points of view in order to explain the striking difference in uniqueness results between 2D and 3D. First, we construct examples of solutions for which the nonlinearity plays a minor role, both in 2D and 3D. Second, we show that, if a solution is rarefied, then its energy is decreasing: hence, rarefaction may be seen as an almost two dimensional assumption. Finally, by exploiting the explicit form of the eigenvectors we provide a numerical explanation of the difficulty in using energy methods for general solutions of the 3D equations.

General solutions’ laws of linear partial differential equations II

This paper uses Z transformations to obtain the general solutions of a large number of second-order, third-order and fourth-order linear partial differential equations for the first time, including one-dimensional inhomogeneous wave equation. Using general solutions, we have obtained plenty of exact solutions of the typical definite solution problems, which proves the important value of general solutions. We propose the Z4 transformation for the first time and initially use it to solve a specific case. We successfully obtain the Fourier series solution using the series general solution of the one-dimensional homogeneous wave equation, which successfully solves a famous unresolved debate in the history of mathematics.

Analyzing the Flexural Performance of Cold-Formed Steel Sigma Section Using ABAQUS Software

Cold-formed steel structures are a type of steel fabrication that is commonly employed in building construction. Before manufacture, they are designed precisely to the appropriate dimensions using the ABAQUS software. Both the strength of the cross-section and distortional buckling determine the load-carrying capability of the section. It was found that thin walls in some cold-formed sections suffered distortional buckling under light loads, and that these elements continued to be strong even after the members buckled. To prevent local buckling, stiffness is offered by the web part. There are several methods for determining the modes and elastic buckling stress. They are finite element analysis, finite strip analysis, and conventional Fourier series solutions. The thickness of the specimen and types of stiffener selection which influence the ultimate strength and deflection should be the issue in the design of the appropriate sigma section. The non-linear numerical analysis of the web-stiffened triangular section was performed using ABAQUS v6.4. It has been demonstrated that sections with height-to-thickness (h/t) ratios have load-bearing capacities. When compared to the lower h/t ratio sections, they demonstrated an improvement in load-carrying capability from 35.13% to 37.2%.

Free vibration analysis of cylindrical sandwich panel with electro-rheological core and FG-GPLRC facing sheets based on First order shear deformation theory referred by Qatu

In this study, free vibration and damping behavior of cylindrical sandwich panel with electro-rheological core and graphene platelets reinforced composite (GPLRC) facing sheets based on first-order shear deformation theory of thick cylindrical shells referred by Qatu (FSDTQ) is investigated. Effective material properties of graphene platelets reinforced composite are determined according to the Halpin–Tsai micromechanical model. The governing equations of motion with required boundary conditions are derived using Hamilton’s principle. Afterward, these equations are solved analytically using the Fourier series solution for simply-supported boundary conditions, and for the other edges boundary conditions, we used the generalized differential quadrature method. The exactness and correctness of the present formulation are validated by comparing the results with those of existing literature. Finally, the influences of various parameters such as the electric fields, geometrically parameters, boundary conditions, the volume fraction of GPL, and GPL distributions pattern on vibration behavior and modal loss factor is examined. According to the results, the influence of the GPL volume fraction and electric field on the frequencies and modal loss factor is significant. This achievement can help design intelligent controlling structures for various applications.

Existence, uniqueness, and stability of Fourier series solutions of stochastic wave equations with cubic nonlinearities in 3D

Existence, uniqueness, and stability of approximate Fourier series solutions u to damped and undamped, semi-linear stochastic wave equations of the form (in Itô integral sense) utt=σ2△u+B(u,ut)+G(u,ut)∂W∂t with cubic nonlinearities B(u,ut)=a1u−a2||u||2u−κut and homogeneous boundary conditions (HBCs) on general 3D cubes D=[0,lx]×[0,ly]×[0,lz] is studied. The driving Q-regular space–time noise W with linear-growth bounded, state-dependent diffusion intensities G(u,ut) is supposed to be general in space (x,y,z)∈D, but white in time t≥0. The analysis is carried out on an appropriate, separable Hilbert space (i.e. the space of Fourier coefficients) of all solutions which are driven by the eigenfunctions of Laplace operator △ subject to HBCs on the 3D domain D. The dynamics of related expected total energy functional e(t) plays a central role in our discussion, depending on diverse parameters such as diffusivity constant σ, damping κ≥0, transport coefficient a1, length parameters lx,ly,lz, diffusion constants br of G and eigenvalues αn,m,li,j,k of related covariance operator Q. Several examples illustrate the feasibility of our approach and explain our major results.

Computing hydrodynamic eigenmodes of channel flow with slip—A highly accurate algorithm

AbstractThe transient start‐up flow solution with slip is a useful tool to verify computational fluid dynamics (CFD) simulations. However, a highly accurate, open‐source black box solution does not seem to be available. Our method provides a fast, automated, and rigorously verified open‐source implementation that can compute the hydrodynamic eigenmodes of a two‐dimensional channel flow beyond the standard floating‐point precision. This allows for a very accurate computation of the corresponding Fourier series solution. We prove that all roots are found in all special cases for the general flow problem with different slip lengths on the channel walls. The numerical results confirm analytically derived asymptotic power laws for the leading hydrodynamic eigenmode and the characteristic timescale in the limiting cases of small and large slip. The code repository including test cases is publicly available (DOI: 10.5281/zenodo.6806351). The Navier slip boundary condition for numerical simulations in OpenFOAM is also publicly available (DOI: 10.5281/zenodo.7037712).

Probing the Miscibility Gap of the Pt–Pd Binary System by Atom Probe Tomography

Abstract To solve the uncertainty of the platinum (Pt)–palladium (Pd) phase diagram, especially the existence of a suggested miscibility gap, atom probe tomography (APT) was used to determine the time evolution of the composition after heat treatment. Due to the extraordinarily slow diffusion in the temperature range of the controversial phase separation, the investigated volume was limited to nano-sized multiple layers deposited by ion beam sputtering (IBS). The evaporated volume was reconstructed from the obtained datasets and the respective diffusion coefficients were determined using the Fourier series solution of the diffusion equation. Beginning with pure Pt and Pd layers annealed at 673, 773, 873, and 973 K, the mixing appears to be purely diffusion controlled in the chosen annealing times, but the state of complete mixing was still not observed. Therefore, extended isothermal annealing sequences at 673 and 773 K with pre-alloyed layers have been carried out. They clearly suggest complete mixing even at the lowest investigated temperatures.

Analytical Solution and Consolidation Analysis of One-dimensional Large Strain Consolidation Differential Equation

Abstract:: According to the universal one-dimensional consolidation equation introduced by Gibson, the governing equation with the excess pore water pressure as the control variable is derived, and the Fourier series solution under the boundary condition of single-sided drainage is deduced in detail by the standard mathematical physical method. It verifies the correspondence between the analytical solution and the numerical solution from a theoretical point of view. Using this analytical solution, the nonlinear distribution of the excess pore pressure along the depth direction is obtained, and the traditional small strain consolidation is compared in terms of the average consolidation degree and the final settlement. Soft soil foundation, large deformation foundation Derive a consolidation equation for soft soils with large deformations using the super-static pore pressure as the control variable Formula derivation, Example analysis Based on Gibson's general equation of consolidation and its theory, the detailed derivation process of differential equations with excess pore water pressure as the control variable is given. According to the example, the image shows the distribution of excess pore pressure with depth, and comparative analysis of large and small strains, If all other conditions are the same, When mv1=1 MPa-1, it can be calculated according to the large and small strains, but when mv1≥3MPa-1, the two errors are large, The calculation must be considered separately.

Comparison of analytical methods and numerical methods in modeling the physical phenomenon of heat conduction with free radiation

The development of mathematical setting for modeling heat conduction phenomena in the presence of radioactive effects is well known. The importance in the study of heat conduction in relation to radioactive effects is relevant in several engineering applications such as combustion, materials science, fluid mechanics and other areas. This research is based on the mathematical model of the heat equation to study the physical phenomenon of heat transfer along a metal bar with slightly insulated sides with the effect of free radiation. To calculate the temperature function that allows modeling the heat transfer process along the bar, the Fourier series solution is constructed step by step, in addition an alternative method of calculating the temperature by using the explicit numerical method is given. The calculation of the temperature along the bar is compared by analytical and numerical method by computing the percentage error and different temperature profiles are plotted to verify the fit of the two approaches. The methods developed throughout the research can be extended to other types of physical phenomena that are useful in related research and in education in engineering subjects such as fluid mechanics and heat transfer.

A complex Fourier series solution for buckling of VAT composite laminates with elastically restrained edges

Variable Angle Tow (VAT) composites have general anisotropic characteristics, and their fiber orientation angle varies continuously along an arbitrary direction. Therefore, the investigation of their buckling performance is quite challenging. Different from the commonly used numerical methods, a new method called complex Fourier series solution is proposed to study the buckling of VAT laminates with elastically restrained edges. First the prebuckling analysis is achieved by introducing Airy’s stress function and expanding it into a complex Fourier series form. And the corresponding mid-plane internal forces with non-uniform distribution are determined. Based on classical laminated plate theory, the buckling analysis of VAT laminate is carried out. The transverse displacement is also expressed as a complex Fourier series. And the critical buckling load and buckling mode shape of VAT laminate are obtained. The accuracy of the presented complex Fourier series solution is verified by comparing with the existing results. The influence of fiber orientation angles, boundary stiffness and boundary conditions on the critical buckling load and buckling mode shape is investigated in numerical examples. Results show that boundary stiffness as well as fiber path have great influence on the critical buckling load and buckling mode shape of VAT laminate. By adjusting the fiber orientation angles and the translational stiffness of the elastically restrained boundary can effectively improve the critical buckling load of VAT laminate. This paper provides a new method for the analysis of general anisotropic laminates with in-plane variable stiffness and the present complex Fourier series solution can also be used as a benchmark for the verification of other numerical results.

Mechanism study and power transmission feature of acoustically stimulated and thermally loaded composite shell structures with double curvature

Because of the modification in environmental conditions, thermal stresses may induce dynamic instability in a composite shell model. Therefore, according to the double Fourier series solution, an analytical strategy is proposed in this study for the first time to acoustically determine the vibration response of a doubly curved composite shell in a thermal environment. The main aim is to show the effect of thermal loads on sound transmission loss (STL). The motion equations of the structure integrated with thermal stresses are derived considering shear deformation shallow shell theory (SDSST) through Hamilton’s principle. Before verifying the outcomes, the convergence of the STL curve is also checked to prove that the obtained results are reliable. Parametric studies are performed to either show the temperature effect on acoustic characteristics or propose a 3D configuration of thermo-acoustic pressures through the system. Based on the temperature variations, it is found that there is an opposite trend between the increase of this term and the STL of structure. This issue also causes to move the dip point of the STL spectrum to lower frequencies. To manifest the fact of these behaviors, the natural frequency ratio against temperature rise is also plotted.

Thin rectangular plates under axial point loading: Accuracy of the classical single Fourier series solution for stresses

The purpose of this article is to clarify certain ambiguities associated with commonly adopted solution strategies of the problem of thin rectangular plates subjected to equal and opposite colinear point loads. The problem is treated as a plane-stress problem where the boundary conditions are prescribed exclusively in terms of tractions. Both an elaborate exact solution and a simplified approximate Single Fourier Series (SFS) solution are employed, shedding light on the accuracy of the latter. This is accomplished, first, by showing that the error implicit in the sine single-Fourier series solution becomes significant for column plates, and, secondly, by re-evaluating the stress distribution at various cross-sections away from the compressed sides, through the application of a truly exact solution to the problem at hand, in which the free-edge conditions are precisely fulfilled. It is shown that the restrictions usually associated with the SFS technique may not be severe and that a wider range of plate shapes with free vertical sides could be amenable to analysis utilizing this simple approximate scheme.