Closed-form Analytical Solution Research Articles

Multiphysics modelling in PyLith: poroelasticity

SUMMARY PyLith, a community, open-source code for modelling quasi-static and dynamic crustal deformation with an emphasis on earthquake faulting, has recently been updated with a flexible multiphysics implementation. We demonstrate the versatility of the multiphysics implementation by extending the code to model fully coupled continuum poromechanics. We verify the newly incorporated physics using standard benchmarks for a porous medium saturated with a slightly compressible fluid. The benchmarks include the 1-D consolidation problem as outlined by Terzaghi, Mandel’s problem for the 2-D case, and Cryer’s problem for the 3-D case. All three benchmarks have been added to the PyLith continuous integration test suite. We compare the closed form analytical solution for each benchmark against solutions generated by our updated code, and lastly, demonstrate that the poroelastic material formulation may be used alongside the existing fault implementation in PyLith.

Analyzing the backbone curve of piecewise-linear non-smooth systems using a generalized bilinear frequency approximation method

Estimating the resonant frequency of non-smooth systems with contacting elements is critical for the design, control, and health monitoring of various mechanical and aerospace structures. The nonlinearity caused by the intermittent contact excludes the use of efficient linear methods; hence, predicting vibration characteristics of these systems becomes computationally challenging. Furthermore, several numerical and analytical approaches that have been developed for analyzing smooth nonlinear systems cannot be applied to analyze non-smooth oscillators. To understand the dynamic properties of non-smooth dynamical systems, the bilinear frequency approximation (BFA) method, a classical analytical tool, was developed to estimate the nonlinear natural frequency of bilinear oscillators by using the linear characteristics of underlying linear subsystems to construct a closed-form analytical solution. However, the traditional BFA method is only applicable to systems in which the gap size between the contacting element is zero, which limits the capability of this efficient analytical technique. In this paper, a generalized BFA method is proposed to extend the capability of the method to capture the nonlinear natural frequency of high-dimensional non-smooth systems where either gaps or prestress exist. The generalized BFA method employs the natural frequencies, mode shapes, and modal amplitudes of the linear subsystems as well as the gap size at the contacting degrees of freedom (DOF) to estimate the nonlinear natural frequencies at different amplitude levels. The backbone curve of non-smooth oscillators can be efficiently constructed using the proposed analytical method. The generalized BFA method is demonstrated on a single-DOF system, a three-DOF system, and a finite element model (FEM) of a cantilever plate with contacting elements. The results show that the proposed method provides an accurate approximation of the backbone curve for the first resonance, which has the most dominant vibration response, with very low computational effort, but a rather poor approximation for higher resonances. The proposed technique can be integrated with high-dimensional FEMs to analyze the nonlinear resonance of complex structures. The performance of the method is validated both numerically and experimentally in this work.

Progress analysis of a multi-recombinative evolution strategy on the highly multimodal Rastrigin function

A first and second order progress rate analysis was conducted for the intermediate multi-recombinative Evolution Strategy (μ/μI,λ)-ES with isotropic scale-invariant mutations on the highly multimodal Rastrigin test function. Closed-form analytic solutions for the progress rates are obtained in the limit of large dimensionality and large populations. The first order results are able to model the one-generation progress including local attraction phenomena. Furthermore, a second order progress rate is derived yielding additional correction terms and further improving the progress model. The obtained results are compared to simulations and show good agreement, even for moderately large populations and dimensionality. The progress rates are applied within a dynamical systems approach, which models the evolution using difference equations. The obtained dynamics are compared to real averaged optimization runs and yield good agreement. The results improve further when dimensionality and population size are increased. Local and global convergence is investigated within given model showing that large mutations are needed to maximize the probability of global convergence, which comes at the expense of efficiency. An outlook regarding future research goals is provided.

Closed-Form Analytical Solutions for the Deflection of Elastic Beams in a Peridynamic Framework

Peridynamics is a continuum theory that operates with non-local deformation measures as well as long-range internal force/moment interactions. The resulting equations are of the integral type, in contrast to the classical theory, which deals with differential equations. The aim of this paper is to analyze peridynamic governing equations for elastic beams. To this end, the strain energy density is formulated as a function of the non-local curvature. By applying the Lagrange principle, the peridynamic equations of motion are derived. Examples of non-local boundary conditions, including simple support, clamped edge, roller clamped edge, and free edge, are presented by introducing the interaction domain. Novel closed-form analytical solutions to integral equations are presented for beams with various boundary conditions, including clamped—simply supported, clamped–clamped, simply supported–roller-clamped, and clamped–roller-clamped beams. Furthermore, different types of loadings, including uniformly distributed load, concentrated force, and concentrated moment, are considered. The results are validated by comparing the derived solutions against solutions to the classical Bernoulli–Euler beam theory. A very good agreement between the non-local and the classical theories is observed for the case of the small horizon sizes, which shows the capability of the derived equations of motion and proposed boundary conditions.

Computation of High-Order Sensitivities of Model Responses to Model Parameters—II: Introducing the Second-Order Adjoint Sensitivity Analysis Methodology for Computing Response Sensitivities to Functions/Features of Parameters

This work introduces a new methodology, which generalizes the extant second-order adjoint sensitivity analysis methodology for computing sensitivities of model responses to primary model parameters. This new methodology enables the computation, with unparalleled efficiency, of second-order sensitivities of responses to functions of uncertain model parameters, including uncertain boundaries and internal interfaces, for linear and/or nonlinear models. Such functions of primary model parameters customarily describe characteristic “features” of the system under consideration, including correlations modeling material properties, flow regimes, etc. The number of such “feature” functions is considerably smaller than the total number of primary model parameters. By enabling the computations of exact expressions of second-order sensitivities of model responses to model “features”, the number of required large-scale adjoint computations is greatly reduced. As shown in this work, obtaining the first- and second-order sensitivities to the primary model parameters from the corresponding response sensitivities to the feature functions can be performed analytically, thereby involving just the respective function/feature of parameters rather than the entire model. By replacing large-scale computations involving the model with relatively trivial computations involving just the feature functions, this new second-order adjoint sensitivity analysis methodology reaches unsurpassed efficiency. The applicability and unparalleled efficiency of this “2nd-Order Function/Feature Adjoint Sensitivity Analysis Methodology” (2nd-FASAM) is illustrated using a paradigm particle transport model that involves feature functions of many parameters, while admitting closed-form analytic solutions. Ongoing work will generalize the mathematical framework of the 2nd-FASAM to enable the computation of arbitrarily high-order sensitivities of model responses to functions/features of model parameters.

Parameterized Linear Power Flow for High Fidelity Voltage Solutions in Distribution Systems

This paper introduces a new model for highly accurate distribution voltage solutions, coined as a parameterized linear power flow model. The proffered model is grounded on a physical model of linear power flow equations, and uses learning-aided parameterization to increase the fidelity of voltage solutions over a wide range of operating points. To this end, the closed-form analytic solution of the parameterization approach is obtained via a Gaussian Process using a deliberately small input sample and without the need for recomputation. The resulting "self-adjusting" parameter is system-specific and controls how accurate the proposed power flow equations are according to loading conditions. Under a certain value of the resulting parameter, the proposed model can fully recover the linearized formulation of a specialized branch flow model for radial distribution systems, the so-called simplified DistFlow model. Numerical examples are provided to illustrate the effectiveness of the proposed model as well as the improvement in solution accuracy for voltage magnitudes over the simplified DistFlow model and several other linear power flow models, at multiple loading levels. Simulations were carried out on six small- and medium-sized test systems.

Stress Relaxation in Bended Viscoelastic Plate with Tension-Compression Asymmetry

The paper presents closed-form analytical solution to the plane-strain problem of stress relaxation in a bended plate with tension-compression asymmetry (TCA) in viscous properties. Reversible and irreversible strains are assumed to be finite. We utilize a linear viscous model with equivalent stress that is piecewise linear function of the principal stresses with TCA parameter. The specific features of the solution are discussed.

An analytical closed-form solution for free vibration and stability analysis of curved sandwich panels made of porous metal-foam core and nanocomposite reinforced face-sheets

This paper aims to obtain an analytical solution for free vibration and stability analysis of shallow curved sandwich panels made of a porous metal-foam core and nanocomposites reinforced sheets resting on a Winkler-Pasternak elastic foundation. In this regard, two nanocomposites, carbon nanotubes (CNTs) and graphene nanoplatelets (GNPs) are considered for reinforcing the face-sheets. The governing equations are developed by considering the transversal shear strains based on the first-order shear deformation theory (FSDT). The obtained system of differential equations representing the dynamic equilibrium and compatibility of the proposed curved sandwich panels are solved analytically, using the principal of Galerkin method. In order to examine the results of developed formulation, some benchmark examples are adopted from the existing literature and the obtained results corresponding to the considered examples are compared with those are presented in the references. Using the achieved numerical results in the framework of a comprehensive parametric study, the effects of various material and geometrical factors and also various boundary conditions on the natural frequencies and buckling loads of the proposed sandwich panels are assessed. By performing the extensive parametric study it was observed that several results corresponding to the free vibration and stability of the proposed sandwich panels can be achieved by the present method without spending much time for computations. On the contrary, in order to achieve the similar results using the numerical methods, a time-consuming process is required which can justify the novelty of the present work.

Analysis of a thick cylindrical FGM pressure vessel with variable parameters using thermoelasticity

Abstract In this study, a closed-form analytical solution is derived to compute the stress formulations for a thick cylindrical pressure vessel made of functionally graded material (FGM) with varying parameters, which are mechanical and thermal boundary conditions. The assumed mechanical boundary condition is the time-dependent pressure acting on the internal surface of the cylinder, while the assumed thermal boundary condition is the transient temperature distribution over the cylinder thickness. The material properties are considered to be graded exponentially in the radial direction, except Poisson’s ratio which is assumed to be constant. The stress and displacement formulations are evaluated using Mathematica software for the uncoupled thermo-mechanical analysis. The results of radial, hoop, and axial stress are plotted at various times for two FGM cylinders, the SS304-Alumina FGM cylinder and the TZM-SIC FGM cylinder, to study the impact of using different materials for the same boundary conditions on the results. The results obtained in this study are beneficial as these contribute to the design and modeling of cylinders that are exposed to time-dependent internal pressure and transient temperature profiles.

Two-dimensional analytical solutions for multi-segmented piezoelectric panels: An EKM approach

A 2-D closed-form analytical solution for the multi-segmented piezoelectric panel has been developed using the extended Kantorovich method (EKM). In this work, a study on the bending behavior of axially segmented piezoelectric panel with different materials have been carried out. This work consider two equally segmented piezoelectric panels made of piezoelectric fiber-reinforced composite (PFRC-2) and lead zirconate titanate (PZT-5A). Furthermore, a particular case of four equal segmented piezoelectric panel is also considered, which imitates an axially graded piezoelectric actuator. The panels are subjected to various boundary conditions and an uniform pressure loading at the top surface. Along the X-axis, interfaces are assumed to be perfectly bonded to satisfy the continuity of displacement and stress at each segment interface. 2-D finite element (FE) solutions using ABAQUS are also obtained to validate displacement and stresses. A good agreement between the analytical and 2-D FE results is observed for all the cases. The current development can be used as a benchmark solution for axially stacked plates or other complex problems.

A numerical investigation of crack behavior near a fixed boundary using singular integral equation and finite element methods

In this paper, we present the method of integral transforms and the Gauss-Chebyshev quadrature methods to solve the problem of a crack parallel to a fixed boundary under remote tension. We derive a system of singular integral equations of the first kind, specific to the problem at hand, which we numerically solve using Gauss-Chebyshev integration. We specialize our results to the problem of a crack in an infinite plate under remote tension, and show that the relative error in our numerically derived solutions is within machine precision of the closed-form analytical solutions. Stress intensity factors are calculated that are in excellent agreement with those derived by others using different methods. We also demonstrate that both the stress intensity factors and normal σyy(x,y) and shear σxy(x,y) stress fields derived via numerical solution of the singular integral equations, compare well with those determined using the commercially available Abaqus finite element code where the crack is modeled using the eXtended Finite Element Method.

Dynamics of fixed-volume pinned films – dealing with a non-self-adjoint thin-film problem

The use of thin liquid films has expanded beyond lubrication and coatings, and into applications in actuators and adaptive optical elements. In contrast to their predecessors, whose dynamics can be typically captured by modelling infinite or periodic films, these applications are characterized by a finite amount of liquid in an impermeable domain. The global mass conservation constraint, together with common boundary conditions (e.g. pinning), create quantitatively and qualitatively different dynamics than those of infinite films. Mathematically, this manifests itself as a non-self-adjoint problem. This work presents a combined theoretical and experimental study for this problem. We provide a time-dependent closed-form analytical solution for the linearized non-self-adjoint system that arises from these boundary conditions. We highlight that, in contrast to self-adjoint problems, here, special care should be given to deriving the adjoint problem to reconstruct the solution based on the eigenfunctions properly. We compare these solutions with those obtained for permeable and periodic boundary conditions, representing common models for self-adjoint thin-film problems. We show that, while the initial dynamics is nearly identical, the boundary conditions eventually affect the film deformation as well as its response time. To experimentally illustrate the dynamics and to validate the theoretical model, we fabricated an experimental set-up that subjects a thin liquid film to a prescribed normal force distribution through dielectrophoresis, and used high-frame-rate digital holography to measure the film deformation in real time. The experiments agree well with the model and confirm that confined films exhibit a different behaviour which could not be predicted by existing models.

On the Interaction of Damage Evolution and Thermal Buckling in Stepped Circular Bilaminates

Abstract The behavior and evolution of stepped circular bi-laminates with edge damage are studied for structures subjected to uniform thermal load. The problem is formulated as a moving intermediate boundaries problem in the calculus of variations, where the boundary of an evolving region of damage emanating from the edge of the smaller substructure, as well as the boundary of a progressing/regressing region of sliding contact adjacent to the intact region of the composite structure are each allowed to vary along with the displacements. This yields the associated transversality conditions that define the locations of the propagating boundaries that correspond to equilibrium configurations of the evolving composite structure, as well as the equations of equilibrium and the associated interior and exterior boundary conditions. Various configurations of contact of the detached segments of the composite structure and the associated behavior are considered, and the influence and progression of contact on the overall evolution of the composite structure are assessed. Closed-form analytical solutions to the geometrically nonlinear problem are obtained, and expressions for the critical buckling load are developed. The explicit forms of the total energy release rate along the delamination front, as well as of the conditions for propagation of the contact zone boundary, are obtained from the analytical solutions and the corresponding transversality conditions. Results of numerical simulations based on the analytical solutions are presented and are seen to unveil a rich evolution process involving contact progression/recession and metamorphosis, buckling, and detachment progression during the prebuckling phase, during sling-shot buckling, and during the postbuckling phase, depending on the material properties of the sublaminates, the geometry of the sublaminates, the initial size of the damage, and the strength of the interfacial bond. Characteristic behavior of damage propagation is found to be quite robust and is seen to include stable propagation, stable followed by unstable progression, arrest, and catastrophic propagation.

An Improved Incremental Procedure for the Ground Reaction Based on Hoek-Brown Failure Criterion in the Tunnel Convergence-Confinement Method

The purpose of this study is to introduce the Hoek–Brown nonlinear failure criterion into the convergence–confinement method (CCM), which is based on the linear failure criterion, and implant it into the ground reaction curve (GRC) in this theory. Based on consistent and rigorous theoretical analysis and according to the mechanical relationships, the principle of equilibrium, the material constitutive law, and the strain/displacement compatibility equation are used to deduce the closed-form analytical solution of the stress/displacement of the rock mass in the plastic region and to complete the simulation and interpretation of the nonlinear behavior of surrounding rock due to the advancing excavation of a circular tunnel. In this paper, confinement loss is regarded as the concept of increment in numerical analysis, and the calculation process and analysis steps of the explicit analysis method (EAM) are proposed. This method can not only realize the calculation of analytical solutions but can also be performed using simple spreadsheets. In addition, to verify the feasibility of this calculation method, this study uses published data to verify the validity of the analytical solution implemented by the incremental procedure and to study the influence of nonlinear failure criteria on the GRC, especially the mechanical behavior at the intrados of the tunnel and the distributions of tress/displacement on the cross-section of the tunnel. A comparison between the published results and the proposed solution in this study reveals consistent and favorable trends.

Harmonic Spectra Analysis of Digital SPWM in VSI With DC Bus Ripple and Dead-Time Effects

This paper presents a closed-form analytical solution to the harmonic spectrum of output voltage of a single-phase H-bridge inverter, and gives the precise harmonic spectrum considering the effects of dead-time and DC bus voltage ripple in a digitally controlled system. Since H-bridge is the basic block in converters, this solution and spectrum can be extended and applied to a variety of pulse width modulation (PWM)-based topologies. The interaction between modulation waveform and triangular during modulation process is fully analyzed from a mathematical perspective, so that the combined effects of digital sampling, dead-time and DC bus voltage ripple on the output voltage are summarized in time and frequency domains. Furthermore, the presented harmonic spectra do not need to judge the current polarity in calculation, which means that the exact amplitude and phase of each output harmonic can be obtained accurately, providing the important theoretical basis for system parameter design and specific harmonic compensation or elimination. The presented analysis results are verified by simulations and experiments.

Resource Allocation of NOMA Communication Systems for Federated Learning

Federated learning is a new communication and computing concept that allows naturally distributed data sets (e.g., as in data acquisition sensors) to be used to train global models, and therefore successfully addresses privacy, power and bandwidth limitations in wireless networks. In this paper, we study the communications problem of latency minimization of a multi-user wireless network used to train a decentralized machine learning model. To facilitate low latency, the wireless stations (WSs) employ non-orthogonal multiple access (NOMA) for simultaneous transmission of local model parameters to the base station, subject to the users’ maximum CPU frequency, maximum transmit power, and maximum available energy. The proposed resource allocation scheme guarantees fair resource sharing among WSs by enforcing only a single WS to spend the maximum allowable energy or transmit at maximum power, whereas the rest of the WSs transmit at lower power and spend less energy. The closed-form analytical solution for the optimal values of resource allocation parameters is used for efficient online implementation of the proposed scheme with low computational complexity.

Non-Newtonian Flow with Heat and Mass Transfer Over a Porous Stretching Sheet Under the Effects of Space Dependent and TGD Heat Sink, Non-Uniform Heat Source and Dissipation of Energy and CGD Mass Diffusion with Suction/Blowing

In the present paper, an analytical study of visco - elastic fluid flow with heat and mass transfer characteristics over a porous stretching sheet has been examined. Heat balance is maintained with space dependent and temperature gradient dependent heat sink/source, viscous dissipation, and non-uniform heat source on the non-Newtonian Walter’s liquid B' model. The mass balance is maintained with chemically reactive species of order I, variable mass diffusivity and concentration gradient dependent mass diffusion. Using similarity transformation technique on the highly non-linear differential equations, several closed form analytical solutions are obtained for non-dimensional temperature and concentration in both PST and PHF cases in the form of confluent hypergeometric (Kummar’s) functions. The effect of permeability parameter, viscoelastic parameter, suction parameter on velocity profiles and various physical fluid situations with different degrees of viscoelasticity, Prandtl number, Eckert number, heat source- sink strength, temperature field are discussed in detail and presented through graphs. Similarly in the mass transfer field the effects of Schmidt number, reaction rate parameters are discussed in detail and presented through graphs.

Micropolar Fluid Flow Through a Porous Stretching/Shrinking Sheet with Mass Transpiration: An Analytical Approach

In this paper, the flow of a micropolar fluid over a stretching or shrinking sheet is investigated under magnetohydrodynamic (MHD) conditions. Such a flow is described by highly nonlinear PDEs. Using the similarity transformation technique, the PDEs governing the flow are reduced to a system of nonlinear ODEs, which further allows a closed-form analytical solution. The effect of the microrotation on the skin friction coefficient, the dimensionless forms of the velocity, and the temperature flow fields in the neighborhood of the stretching or shrinking sheet are discussed for various combinations of the dimensionless parameters. The numerical results reveal that the micropolar flow may accelerate or deaccelerate depending upon the numerical values of the mass transpiration and the permeability of the porous sheet. An increase in the tangential and the angular flow velocities is found to occur with an increase in the microrotation. Further, it is observed that the increase in the microrotation increases the skin friction coefficient.

Inventory management for stockout-based substitutable products under centralised and competitive settings

Inventory planning in fashion markets is highly challenging, owing to uncertain demand; yet, in making inventory decisions, retailers may be able to capitalise on high substitutability between products. This research develops single-period inventory-management models describing a market with two substitutable products, under stockout-based substitution; i.e. when a customer’s preferred product is out-of-stock, s/he may choose to purchase the substitute. Two settings are considered: centralised (a single retailer who sells both products) and competitive (two retailers, each selling one product). For each setting, we derive closed-form analytical solutions for the inventory levels that maximise expected profit. The model is further enriched with sales data from an online apparel retailer offering substitutable products (a sneaker in different colours), and we analyse the sensitivity of the optimal inventory levels and profits to parameter values. Key findings include the following: (i) Under competitive conditions, both retailers always order positive inventory so as not to lose customers. However, in a single-retailer setting, there are situations in which the retailer orders inventory for only one product. (ii) The optimal inventory levels and corresponding profits are highly sensitive to consumers’ willingness to substitute between products. These findings provide concrete insights that can guide fashion brands’ inventory-management decisions.

The inverse Mellin transform via analytic continuation

We present a method to calculate the x-space expressions of massless or massive operator matrix elements in QCD and QED containing local composite operator insertions, depending on the discrete Mellin index N, directly, without computing the Mellin-space expressions in explicit form analytically. Here N belongs either to the even or odd positive integers. The method is based on the resummation of the operators into effective propagators and relies on an analytic continuation between two continuous variables. We apply it to iterated integrals as well as to the more general case of iterated non-iterative integrals, generalizing the former ones. The x-space expressions are needed to derive the small-x behaviour of the respective quantities, which usually cannot be accessed in N-space. We illustrate the method for different (iterated) alphabets, including non-iterative 2F1 and elliptic structures, as examples. These structures occur in different massless and massive three-loop calculations. Likewise the method applies even to the analytic closed form solutions of more general cases of differential equations which do not factorize into first-order factors.