Fourier Series Representation Research Articles

A note on existence and uniqueness of solutions for a 2D bioheat problem

We consider a 2D bioheat model on a domain with curvilinear polygonal boundary and boundary conditions involving heat transfer between blood vessels and tissue. Based on elliptic regularity on polygons and semigroup theory, we obtain a Fourier series representation for the solution in settings. The results become useful in that they provide theoretical support for numerical approaches recently published in the literature.

Locally-exact homogenization of unidirectional composites with coated or hollow reinforcement

Effects of coated and hollow reinforcement on homogenized moduli and stress fields in unidirectional composites are investigated using elasticity-based homogenization theory for periodic materials with hexagonal and tetragonal symmetries extended to accommodate coatings. The theory employs Fourier series representations for fiber, coating and matrix displacement fields in the cylindrical coordinate system that satisfy exactly equilibrium equations and continuity conditions in the interior of the unit cell. The inseparable exterior problem requires satisfaction of periodicity conditions efficiently accomplished using previously introduced balanced variational principle which ensures rapid displacement solution convergence in the presence of thick or very thin coatings with relatively few harmonic terms. The solution's stability facilitates rapid identification of coatings' impact on homogenized moduli and local fields in wide ranges of fiber volume fraction and coating thickness. Elastic response of unidirectional composites reinforced by hollow fibers may be obtained without specializing the theory's framework by appropriately adjusting input parameters. This is illustrated using alumina nanotubes as reinforcement, revealing new results of interest in the design of multifunctional porous materials. Equally important, the theory's analytical framework requires minimal effort in constructing input data file that defines the unit cell problem, facilitating use by researchers with little mechanics exposure.

Analysis of thermoelastic stress-concentration around oblate cavities in three-dimensional generally anisotropic bodies by the boundary element method

In this article, stress concentrations around cavities in 3D generally anisotropic bodies are investigated using the boundary element method (BEM), where the associated volume integral is analytically transformed to the boundary. All derivations are based upon the Fourier-series representations of the fundamental solutions, including Green's function of displacements and its derivatives. This approach of analytical transformation has fully restored the BEM's distinctive notion that only the boundary needs to be discretized. The goal of the present work is to investigate the thermoelastic stress-concentration around cavities in 3D anisotropic bodies by use of the analytically transformed boundary integral equation (BIE). The work has fully recovered the BEM's nature of boundary discretization for treating 3D generally anisotropic thermoelasticity. This is the first implemented work that successfully treats thermoelastic problems for 3D generally anisotropic solids by the analytically transformed BIE. In the paper, Interesting phenomena are observed from the analyses of stress concentrations around oblate cavities and some discussions are made.

REPRESENTATION OF GRADIENTS OF A SCALAR FIELD ON THE SPHERE USING A 2D FOURIER EXPRESSION

Abstract. Representation of data on the sphere is conventionally done using spherical harmonics. Making use of the Fourier series of the Legendre function in the SH representation results in a 2D Fourier expression. So far the 2D Fourier series representation on the sphere has been confined to a scalar field like geopotential or relief data. We show that if one views the 2D Fourier formulation as a representation in a rotated frame, instead of the original Earth-fixed frame, one can easily generalize the representation to any gradient of the scalar field. Indeed, the gradient and the scalar field itself are simply linked in the spectral domain using spectral transfers. We provide the spectral transfers of the first-, second- and third-order gradients of a scalar field in a local frame. Using three numerical examples based on gravity and geometrical quantities, we show the applicability of the presented formulation.

Locally-exact homogenization theory for transversely isotropic unidirectional composites

The locally-exact homogenization theory for unidirectional composites with square periodicity and isotropic phases proposed by Drago and Pindera [18] is extended to architectures with hexagonal symmetry and transversely isotropic phases. The theory employs Fourier series representation for the displacement fields in the fiber and matrix phases in the cylindrical coordinate system that satisfies exactly the equilibrium equations and continuity conditions in the unit cell's interior. The inseparable exterior problem involves satisfaction of periodicity conditions for the hexagonal unit cell geometry demonstrated herein to be readily achievable using the previously introduced balanced variational principle for square geometries. This variational principle plays a key role in the employed unit cell solution, ensuring rapid convergence of the Fourier series coefficients with relatively few harmonic terms, yielding converged homogenized moduli and local stress fields with little computational effort. The solution's stability is illustrated using the dilute case which is shown to reduce to the Eshelby solution regardless of the employed number of harmonic terms. Comparison with published results and predictions of a finite-volume based homogenization in a wide fiber volume range and different fiber/matrix modulus contrast validates the approach's accuracy, and its utility is demonstrated through rapid local stress recovery in a multi-scale application. This extension completes the development of the theory for three important classes of unidirectional reinforcement arrays, thereby providing an efficient alternative to finite-element based homogenization techniques or approximate micromechanical schemes, as well as an efficient standard against which other methods may be compared.

Intrinsic random functions and universal kriging on the circle

Intrinsic random functions (IRF) provide a versatile approach when the assumption of second-order stationarity is not met. Here, we develop the IRF theory on the circle with its universal kriging application. Unlike IRF in Euclidean spaces, where differential operations are used to achieve stationarity, our result shows that low-frequency truncation of the Fourier series representation of the IRF is required for such processes on the circle. All of these features and developments are presented through the theory of reproducing kernel Hilbert space. In addition, the connection between kriging and splines is also established, demonstrating their equivalence on the circle.

Numerical Solvability and Solution of an Inverse Problem Related to the Gibbs Phenomenon

We report on the inverse problem for the truncated Fourier series representation of fx∈BV-L,L in a form with a quadratic degeneracy, revealing the existence of the Gibbs-Wilbraham phenomenon. A new distribution-theoretic proof is proposed for this phenomenon. The paper studies moreover the iterative numerical solvability and solution of this inverse problem near discontinuities of fx.

Improving Space Object Catalog Maintenance Through Advances in Solar Radiation Pressure Modeling

This paper investigates the weaknesses of using the cannonball model to represent the solar radiation pressure force on an object in an orbit determination process, and it presents a number of alternative models that greatly improve the orbit determination performance. These weaknesses are rooted in the fact that the cannonball model is not a good representation of the true solar radiation pressure force acting on an arbitrary object. Using an erroneous force model results in poor estimates, inaccurate trajectory propagation, unrealistic covariances, and the inability to fit long and/or dense arcs of data. The alternative models presented are derived from a Fourier series representation of the solar radiation pressure force. The simplest instantiation of this model requires only two more parameters to be estimated, however, this results in orders of magnitude improvements in tracking accuracy. This improvement is illustrated through numerical examples of a discarded upper stage in a geosynchronous transfer orbit, and more drastically for a piece of high-area-to-mass ratio debris in a near-geosynchronous orbit. Implementation of improved solar radiation pressure models in this manner will alleviate track correlation, object identification, and sensor tasking issues that plague current catalog maintenance due to the inaccurate standard dynamic model.

Analytical solution of lipid membrane morphology subjected to boundary forces on the edges of rectangular membranes

We develop a complete analytical solution predicting the deformation of rectangular lipid membranes resulting from boundary forces acting on the perimeter of the membrane. The shape equation describing the equilibrium state of a lipid membrane is taken from the classical Helfrich model. A linearized version of the shape equation describing membrane morphology (within the Monge representation) is obtained via a limit of superposed incremental deformations. We obtain a complete analytical solution by reducing the corresponding problem to a single partial differential equation and by using Fourier series representations for various types of boundary forces. The solution obtained predicts smooth morphological transition over the domain of interest. Finally, we note that the methods used in our analysis are not restricted to the particular type of boundary conditions considered here and can accommodate a wide class of practical and important edge conditions.

A series solution for the vibrations of composite laminated deep curved beams with general boundaries

In this paper, a series solution is derived for the vibration analysis of composite laminated deep curved beams with general boundary conditions. The effects of shear deformation, inertia rotary and deepness term are considered in the formulation. Under the current framework, the governing equations and the related boundary equations are obtained via the Hamilton’s principle. And each of beam displacements, regardless of boundary conditions, is expanded as a modified Fourier series composed of a standard cosine Fourier series and certain supplementary terms introduced to remove the potential discontinuities at the ends, thus ensure and accelerate the convergence of the series representation. The characteristic equations are then derived directly in an exact sense by solving the equations of motion in matrix form by combining the associated boundary equations and the modified Fourier series representation. The convergence and accuracy of the solution are tested and validated by several numerical cases against the results available in the literature, with excellent agreements obtained. A systematic parametric study is also performed regarding the effects of shear deformation and inertia rotary, deepness term, boundary conditions, lamination schemes, material and geometrical parameters. Finally, several numerical results of composite laminated deep and shallow curved beams with different geometry dimensions are presented for various boundary conditions and lamination schemes, which may serve as benchmark solutions for the future researches in this field.

Research on Initial Geometric Deviation Description for Numerical Simulation of Cylindrical Shells under External Pressure

The simulation result of cylindrical shells under external pressure is influenced greatly by different initial geometric deviation. Two forms of initial geometric deviations i.e., the first-order buckling mode of shell and the Fourier series representation, are briefly introduced. A simplified method of Fourier series is developed according to circumferential wave l(2-8), initial phase angle Φ12-Φ18 and 5 groups initial geometric deviations data. According to the basic dimension, maximum initial geometric deviation, elastic modulus and yield strength of cylindrical shells in the existing reference, simplified Fourier series and first-order buckling mode method are applied to describe the initial geometric deviations of cylindrical shells in the double nonlinear buckling simulation, bilinear material model is adopted to the constitutive relation of materials. The results are discussed and the values regarding the buckling pressure obtained by the simulation are compared with those from experiments reported in reference. The results show that the values of buckling pressure obtained by first-order buckling mode method are generally smaller than the experimental values, and the results obtained by the simplified Fourier series method are in good agreement with the experimental values in the reference. This illustrates that the initial geometric deviations of cylindrical shells can be better expressed by the simplified Fourier series.

Concordancing in ESP Class Rooms

The effect of time periodic body force (or g-jitter or gravity modulation) on the onset of Rayleigh-Bnard electro-convention in a micropolar fluid layer is investigated by making linear and non-linear stability analysis. The stability of the horizontal fluid layer heated from below is examined by assuming time periodic body acceleration. This normally occurs in satellites and in vehicles connected with micro gravity simulation studies. A linear and non-linear analysis is performed to show that gravity modulation can significantly affect the stability limits of the system. The linear theory is based on normal mode analysis and perturbation method. Small amplitude of modulation is used to compute the critical Rayleigh number and wave number. The shift in the critical Rayleigh number is calculated as a function of frequency of modulation. The non-linear analysis is based on the truncated Fourier series representation. The resulting non-autonomous Lorenz model is solved numerically to quantify the heat transport. It is observed that the gravity modulation leads to delayed convection and reduced heat transport.

Linear and Weakly Non-Linear Analyses of Gravity Modulation and Electric Field on the onset of Rayleigh-Bénard Convection in a Micropolar Fluid

The effect of time periodic body force (or g-jitter or gravity modulation) on the onset of Rayleigh-Benard electro-convention in a micropolar fluid layer is investigated by making linear and non-linear stability analysis. The stability of the horizontal fluid layer heated from below is examined by assuming time periodic body acceleration. This normally occurs in satellites and in vehicles connected with micro gravity simulation studies. A linear and non-linear analysis is performed to show that gravity modulation can significantly affect the stability limits of the system. The linear theory is based on normal mode analysis and perturbation method. Small amplitude of modulation is used to compute the critical Rayleigh number and wave number. The shift in the critical Rayleigh number is calculated as a function of frequency of modulation. The non-linear analysis is based on the truncated Fourier series representation. The resulting non-autonomous Lorenz model is solved numerically to quantify the heat transport. It is observed that the gravity modulation leads to delayed convection and reduced heat transport.

Double diffusive convection in a couple stress fluid saturated porous layer with internal heat source

Abstract The onset of double diffusive convection in a couple stress fluid saturated porous layer with an internal heat source is studied using linear and weak nonlinear stability analyses. The linear analysis is based on the classical normal mode technique. The modified Darcy equation that includes the time derivative term is used to model the momentum equation. In the present context, double diffusive convection is of particular interest in the study of extraction of metals from ores where a mushy layer is formed during solidification of a metallic alloy. Further, internal heating of the system is used as an external means to influence the transport process, thereby controlling the quality and structure of the resulting solid. The expressions for stationary, oscillatory and finite-amplitude Rayleigh number are obtained as a function of governing parameters such as internal Rayleigh number, couple stress parameter, solute Rayleigh number, Darcy-Prandtl number, normalized porosity and Lewis number and their effects on the stability of a system are shown graphically. It is observed that the onset of both stationary and oscillatory convection is advanced by the internal Rayleigh number. The nonlinear analysis is based on the truncated representation of Fourier series which provides the quantification of heat and mass transfer. Heat transport is reinforced while mass transport is suppressed by the internal Rayleigh number. The transient behavior of Nusselt and Sherwood numbers is studied by solving numerically a fifth order Lorenz type system using Runge–Kutta method. Some known results are recovered as the particular cases of the present study.

Radial eigenfunctions for the game-theoretic $p$-Laplacian on a ball

The normalized or game-theoretic $p$-Laplacian operator given by $$ -\Delta_p^Nu:=-\frac{1}{p}|\nabla u|^{2-p}\Delta_p(u) $$ for $p\in(1,\infty)$ with $\Delta_pu=\rm{div}(|\nabla u|^{p-2}\nabla u)$ has no apparent variational structure. Showing the existence of a first (positive) eigenvalue of this fully nonlinear operator requires heavy machinery as in [6]. If it is restricted to the class of radial functions, however, the normalized $p$-Laplacian transforms into a linear Sturm-Liouville operator. We investigate radial eigenfunctions to this operator under homogeneous Dirichlet boundary conditions and come up with an explicit complete orthonormal system of Bessel functions in a suitably weighted $L^2$-space. This allows us to give a Fourier-series representation for radial solutions to the corresponding evolution equation $u_t-\Delta_p^Nu=0$.

Analytical transformation of the volume integral in the boundary integral equation for 3D anisotropic elastostatics involving body force

In the boundary element method (BEM), it is well known that the presence of body force shall give rise to an additional volume integral that conventionally requires domain discretization for numerical computations. To restore the BEM’s distinctive notion of boundary discretization, the present work analytically transforms the volume integral to surface ones for the body-force effect in the 3D anisotropic elasticity. On applying Green’s Theorem, new fundamental solutions with explicit forms of Fourier series are introduced to facilitate the volume-to-surface transformation. The coefficients of the Fourier-series representations are determined by solving a banded matrix formulated from integrations of the constrained equation. Of no doubt, such an approach has fully restored the boundary element method as a truly boundary solution technique for analyzing 3D anisotropic elasticity involving body force. At the end, numerical verifications of the volume-to-surface integral transformation are presented. Also, such an approach has been implemented in an existing BEM code. For demonstrating the implementation, numerical examples are presented with comparisons with ANSYS analysis. To the author’s knowledge, this is the first work in the open literature that reports the successful transformation for 3D anisotropic elasticity.

Generalized Representation of Electric Fields in Sheet Beam Klystron Gaps

Kosmahl and Branch's derivation for the electric field in a round beam gap is closely followed to derive the electric field for a sheet beam klystron gap. The wider of the two transverse dimensions of the gap is taken to be infinite in extent and the field is derived based on an approximation of the gap field at the drift tube edge. The electric field equations are generalized using a Fourier series representation of the gap field at the drift tube edge. The analytical results are compared with the numerical computations.

Deterministic and band-limited stochastic energy harvesting from uniaxial excitation of a multilayer piezoelectric stack

Deterministic and band-limited stochastic energy harvesting scenarios using a multilayer piezoelectric stack configuration are investigated for uniaxial dynamic pressure loading. The motivation for exploring this off-resonant energy harvesting problem derives from typical civil infrastructure systems subjected to dynamic compressive forces in deterministic or stochastic forms due to vehicular or human loads, among other examples of compressive loading. Modeling of vibrational energy harvesters in the existing literature has been mostly focused on deterministic forms of mechanical vibration as in the typical case of harmonic excitation, while the efforts on stochastic energy harvesting have thus far considered second-order systems such as piezoelectric cantilevers. In this paper, we present electromechanical modeling, analytical and numerical solutions, and experimental validations of piezoelectric energy harvesting from harmonic, periodic, and band-limited stochastic excitation of a multilayer piezoelectric stack under axial compressive loading in the off-resonant low-frequency range. The deterministic problem employs the voltage output-to-pressure input frequency response function of the harvester for a given electrical load, which is also extended to periodic excitation. The analytical stochastic electromechanical solution employs the power spectral density of band-limited stochastic excitation to predict the expected value of the power output. The first one of the two numerical solution methods uses the Fourier series representation of the excitation history to solve the resulting ordinary differential equation, while the second method employs an Euler–Maruyama scheme to directly solve the governing electromechanical stochastic differential equation. The electromechanical models are validated through several experiments for a multilayer PZT-5H stack under harmonic and band-limited stochastic excitations at different pressure levels. The figure of merit is also extracted for this particular energy harvesting problem to choose the optimal material. Soft piezoelectric ceramics (e.g. PZT-5H and PZT-5A) offer larger power output as compared to hard ceramics (e.g. PZT-8), and likewise, soft single crystals (e.g. PMN-PT and PMN-PZT) produce larger power as compared to their hard counterparts (e.g. PMN-PZT-Mn); and furthermore, single crystals (e.g. PMN-PT and PMN-PZT) generate more power than standard ceramics (e.g. PZT-5H and PZT-5A) for low-frequency, off-resonant excitation of piezoelectric stacks.

Determination of Soil Hydraulic Parameters with CyclicIrrigation Tests

A method for determining soil hydraulic parameters based on periodic point source solutions of the linearized Richards equation is proposed. Closed‐form solutions were derived for buried and surface point sources with a sinusoidally varying flux. These solutions describe the dependence of the matric flux potential (MFP) amplitude and phase shift on the distance from the point sources, frequency of source flux alternations, and soil hydraulic parameters. The Fourier series representation of square waves was used to extend the point source solutions for sinusoidally varying flux to square‐wave cyclic inputs of water with a 50% duty cycle—a preferable operating mode for field experiments. Close to the sources, the amplitudes and shape of MFP waves from a step input were more pronounced than those from a sinusoidal input, but the difference diminished as distance increased. The proposed method involves recording the pressure head variations with a continuously reading tensiometer installed below a pressure‐compensating emitter controlled by an irrigation computer. Among 16 cyclic step‐input irrigation tests, one series involved pressure‐head measurements at a single depth and varied water‐pulse durations; the second involved pressure‐head measurements at various depths with a fixed water‐pulse duration. The soil saturated hydraulic conductivity, the coefficient α of Gardner's conductivity model, and the coefficient k of the linearized unsteady‐flow equation were estimated by matching the periodic solution for cyclic step inputs to the measured pressure‐head data with a two‐step inversion procedure: an amplitude‐shift (AS) estimation procedure followed by the Levenberg–Marquardt (LM) optimization algorithm. The AS procedure was effective, and the LM optimization yielded only minor improvements. Our main findings are: (i) the periodic solution for step inputs yields reasonably good predictions of pressure‐head variations measured at various depths below an emitter for different periods of water application; and (ii) the proposed point‐source method seems to yield consistent and reliable estimates of the three soil hydraulic parameters.

Linear and Nonlinear Stability Analysis of Double Diffusive Convection in a Maxwell Fluid Saturated Porous Layer with Internal Heat Source

The onset of double diffusive convection is investigated in a Maxwell fluid saturated porous layer with internal heat source. The modified Darcy law for the Maxwell fluid is used to model the momentum equation of the system, and the criterion for the onset of the convection is established through the linear and nonlinear stability analyses. The linear analysis is obtained using the normal mode technique, and the nonlinear analysis of the system is studied with the help of truncated representation of Fourier series. The effects of internal Rayleigh number, stress relaxation parameter, normalized porosity, Lewis number, Vadasz number and solute Rayleigh number on the stationary, and oscillatory and weak nonlinear convection of the system are shown numerically and graphically. The effects of various parameters on transient heat and mass transfer are also discussed and presented analytically and graphically.