Numerical Laplace Inversion Research Articles

Piezothermoelastic interaction in piezoelectric rods induced by a timed laser pulse without energy dissipation

The main aim of this article is to study the piezo-thermoelastic interaction in piezoelectric rods without energy dissipation due to a laser heat source with a timed pulse. Laplace transforms are used to represent the analytical solutions in the transformations domain. The governing equations of piezo-thermoelasticity in the Green and Naghdi model (GNII model without energy dissipation) in the presence of a laser heat source with a timed pulse are solved exactly in the Laplace domain, calculating the temperature, the stress, the displacement, and the electric potential. Numerical Laplace inversion is then used to find the general solution of the variables under study, which are then graphically shown.

Generalized piezothermoelastic interactions in a piezoelectric rod subjected to pulse heat flux

Abstract This work investigates, using the Laplace transforms, the influence of thermal relaxation time in the piezo-thermoelastic rod under pulse heat flux. For the piezoelectric medium, the generalized piezothermoelastic fundamental equations are developed. The analytical solutions are expressed in the transformation domain using Laplace transforms. Laplace transforms are presented to solve the problem’s governing equations, removing the time impact and yielding analytical solutions for the temperature, electric field, displacement, and stresses in the Laplace domain. The time domain solutions of the variables under consideration are then found using numerical Laplace inversion and visually shown. The effects of the thermal time, pulse heating flux characteristic time, and constant heat flux are studied in a piezoelectric thermoelastic medium. The figures show that the thermal time, pulse heating flux characteristic time, and constant heat flux play significant roles in determining the values of all physical quantities.

Exact Simulation of Quadratic Intensity Models

We develop efficient algorithms of exact simulation for quadratic stochastic intensity models that have become increasingly popular for modeling events arrivals, especially in economics, finance, and insurance. They have huge potential to be applied to many other areas such as operations management, queueing science, biostatistics, and epidemiology. Our algorithms are developed by the principle of exact distributional decomposition, which lies in a fully analytical expression for the joint Laplace transform of quadratic process and its integral newly derived in this paper. They do not involve any numerical Laplace inversion, have been validated by extensive numerical experiments, and substantially outperform all existing alternatives in the literature. Moreover, our algorithms are extendable to multidimensional point processes and beyond Cox processes to additionally incorporate two-sided random jumps with arbitrarily distributed sizes in the intensity for capturing self-exciting and self-correcting effects in event arrivals. Applications to portfolio loss modeling are provided to demonstrate the applicability and flexibility of our algorithms. History: Accepted by Bruno Tuffin, Area Editor for Simulation. Funding: This work was supported by the Beijing University of Posts and Telecommunications [Grant 2022RC58], the Shanghai University of Finance and Economics [Grant 2020110930], the National Natural Science Foundation of China [Grant 71401147], and Graduate Innovation Fund of Shanghai University of Finance and Economics [CXJJ-2023-387]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0323 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0323 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Decentralized fractional-order PIλ control applied to a TITO distillation column: a comparative study with PID and MPC

This paper investigates the use of fractional-order PIλ (FO-PI) controllers and compares their performance with classical integer-order PID (IO-PID) and fractional-order PID (FO-PID) controllers for a two-input, two-output (TITO) distillation column. Additionally, a comparative study between FO-PI and model predictive control (MPC) is conducted on the same multivariable system. While fractional control in multivariable systems has been explored, evaluations of global performance metrics—such as integral square error (ISE), integral absolute value of error (IAE), integral time weighted absolute error (ITAE), integral time weighted square error (ITSE), and integral of manipulated variables (IU)—compared to IO-PID and MPC are still scarce. This work addresses this gap by examining these metrics alongside the steps necessary for fractional control design, including Laplace numerical inversion and controller tuning. The results demonstrate that the FO-PI controller shows superior performance compared to FO/IO-PID and MPC controllers. The global relative gain indexes show improvements, with FO-PI achieving gains of 36.9% to 59.9% compared to FO/IO-PID, and a reduction of 19% to 44.1% in global index compared to MPC. The distillation column dynamics were modeled using transfer functions, allowing for a detailed comparison of control strategies. The studies conducted in this work indicate that fractional-order controllers (FO-PI) show promise in controlling the dynamics of the TITO distillation column, as evidenced by simulations and comparison with integer-order (IO-PID) and fractional-order (FO-PID) control strategies, as well as with MPC control. The study highlights the potential benefits of using fractional controllers for a classical multivariable system, allowing to advance the state of the art of this type of control approach.

Generalized Finite Integration Method with Laplace transform for European option pricing under Black–Scholes and Heston models

In this paper, we combine a recently developed Generalized Finite Integration Method (GFIM) with Laplace transform technique for pricing options under the Black Scholes model and Heston model respectively. Instead of using traditional time-stepping process, we first perform Laplace transform on the governing equation and boundary conditions to remove the temporal derivatives. The Generalized Finite Integration Method is then exploited to handle the spatial differential operators in the transformed space. From numerical Laplace inversion algorithm, we restore the required time-dependent option price. For verification, several option pricing models governed by one-dimensional Black–Scholes equation and two-dimensional extended Heston equation are constructed to demonstrate the efficiency and feasibility of the proposed approach.

Theoretical analysis and engineering application of controllable shock wave technology for enhancing coalbed methane in soft and low-permeability coal seams

Coalbed methane (CBM) is a significant factor in triggering coal and gas outburst disaster, while also serving as a clean fuel. With the increasing depth of mining operations, coal seams that exhibit high levels of gas content and low permeability have become increasingly prevalent. While controllable shockwave (CSW) technology has proven effective in enhancing CBM in laboratory settings, there is a lack of reports on its field applications in soft and low-permeability coal seams. This study establishes the governing equations for stress waves induced by CSW. Laplace numerical inversion was employed to analyse the dynamic response of the coal seam during CSW antireflection. Additionally, quantitative calculations were performed for the crushed zone, fracture zone, and effective CSW influence range, which guided the selection of field test parameters. The results of the field test unveiled a substantial improvement in the gas permeability coefficient, the average rate of pure methane flowrate, and the mean gas flowrate within a 10 m radius of the antireflection borehole. These enhancements were notable, showing increases of 3 times, 13.72 times, and 11.48 times, respectively. Furthermore, the field test performed on the CSW antireflection gas extraction hole cluster demonstrated a noticeable improvement in CBM extraction. After antireflection, the maximum peak gas concentration and maximum peak pure methane flow reached 71.2% and 2.59 m3/min, respectively. These findings will offer valuable guidance for the application of CSW antireflection technology in soft and low-permeability coal seams.

Dynamic Response of Infinite Beam Resting on a Fractional Pasternak Viscoelastic Foundation Subjected to Moving Load

In this paper, the dynamic response of an infinite Euler beam that was mounted on a fractional-order Pasternak viscoelastic foundation subjected to a moving point load was investigated. An analytical solution to the problem was derived using Fourier and Laplace transforms. Numerical results obtained by numerical Laplace inversion were analyzed to explore the impact of various parameters on the system’s response. The findings indicated that increasing system damping led to a decrease in maximum deflection and a more visible deformation hysteresis with an increase in fractional derivative orders. Additionally, all parameters of the foundation and shear layer were observed to have a significant effect on the deflection. The study confirmed that the fractional-order model predicted damping and dynamic deflection more accurately than the conventional integer-order foundation model. The research contributed to the understanding of the behavior of Euler beams mounted on viscoelastic foundations and provided valuable insights into the design of such systems.

An Elementary Formula for the Initial Relaxation Modulus from the Creep Compliance for Asphalt Mixtures.

The conversion between the relaxation modulus and creep compliance is a traditional research topic in viscoelastic materials. Generally, different methods have been used to solve the numerical solution based on convolution theory. However, the initial relaxation modulus (relaxation modulus at t = 0) has been difficult to obtain. This paper aimed to propose a fast calculation method to derive the initial relaxation modulus from the creep compliance. First, three groups of uniaxial static creep tests of asphalt mixtures were conducted to determine the creep compliance of the experimental data. Then, the calculation of the initial relaxation modulus from the creep compliance by three inversion methods (midpoint method, approximate method, and Laplace numerical inversion method) was evaluated. The results indicate that approximate method and Laplace numerical inversion method cannot calculate the initial relaxation modulus value, and the calculation results of the midpoint method can only approach the exact value infinitely, for which calculating the relaxation modulus at 0.0005 s requires 2000 s. The results can only approach the exact value infinitely and take a lot of computing time. Finally, a fast calculation method for the initial relaxation modulus is proposed and verified by Laplace initial value theorem, and this method can directly derive a simple expression for calculating the initial relaxation modulus without requiring computational time. The proposed calculation methods of the initial relaxation modulus for various viscoelastic models were then put forward. The research results provide an effective tool for obtaining the initial relaxation modulus accurately.

Analysis of Liquid Chromatography Considering a Linear Single-Component Heterogeneous-Type Reactive General Rate Model.

This paper elaborates on the significance of liquid chromatography for a single-component reactive linear general rate model. The model equations consist of a set of two coupled partial differential equations, which include diffusion, interfacial mass transfer, axial dispersion, external and intraparticle pore diffusivity, and heterogeneous chemical reaction of the first order with two sets of boundary conditions. The model equations are solved by the Laplace transformation. The actual time domain solution is obtained by numerical Laplace inversion, as analytical inversion cannot be obtained. The graphical sketch of different physical parameters is presented to analyze the dynamics of the elution profiles. The result indicates that the chromatographic reactor works more efficiently on increasing the value of the heterogeneous-type first-order reaction. To check the analytical results, a second-order high-resolution finite volume scheme is used. Both results are in good agreement and indicate the correctness of the numerical scheme. The current work is also compared with the previously available numerical schemes, which shows that the proposed numerical scheme is better for elaborating the chromatographic reactor performance. A comparison table is also presented to compute error analysis and computational run time for analyzing the efficiency of the reactor. A graphical sketch of the numerical temporal moment analysis is also presented, which gives significant information about the performance and the shape of the concentration profiles.

Transient non-Fourier thermal interactions of two parallel cracks in porous metal foam

Experimental evidence shows heat conduction in the porous media would be non-Fourier under specific circumstances. The novelty of this research lies in considerations of the non-Fourier effect and the interactions of two parallel thermally insulated cracks in the porous metal foam. The transient propagation characteristics are examined by utilizing the Dual-phase-lag model, which incorporates the certain time required to establish local thermal equilibrium between the solid framework and surrounding pores. Fourier transform and Laplace transform are employed to solve the mathematical model simulating a porous strip subjected to abrupt thermal shock. After reducing the problem to two groups of singular integral equations, the transient temperature distributions and heat flux intensity factors are determined with the help of numerical Laplace inversion. Numerical calculations are carried out to evaluate the impacts of crack positions, the relative density, phase lags, and two crack lengths on the thermal characteristics, which would contribute to a thorough understanding of the ultrafast process of porous materials operating in extreme conditions.

Generalized Thermoelastic Heat Conduction Model Involving Three Different Fractional Operators

Abstract The purpose of this paper is to introduce a new time-fractional heat conduction model with three-phase-lags and three distinct fractional-order derivatives. We investigate the introduced model in the situation of an isotropic and homogeneous solid sphere. The exterior of the sphere is exposed to a thermal shock and a decaying heat generation rate. We recuperate some earlier thermoelasticity models as particular cases from the proposed model. Moreover, the effects of different fractional thermoelastic models and the effect of instant time on the physical variables of the medium are studied. We obtain the numerical solutions for the various physical fields using a numerical Laplace inversion technique. We represent the obtained results graphically and discuss them. Physical views presented in this article may be useful for the design of new materials, bio-heat transfer mechanisms between tissues and other scientific domains.

Two‐region flow caused by pumping at a partial penetration well in a leaky confined aquifer

AbstractAn analytical model for depicting two‐region flow caused by constant‐rate pumping at a partially penetrating well in a leaky confined aquifer is established. The two‐region flow is described by the radial Izbash non‐Darcy flow in the vicinity of the abstraction well and Darcy flow away from the abstraction well in the pumped aquifer. The model considers the effects of aquitard storage, wellbore storage, and wellbore skin. The semi‐analytical solutions of drawdown are developed by means of a linearization procedure combined with Laplace transform and separation of variables. The time‐domain drawdowns are then obtained by using the Stehfest method for numerical Laplace inversion. The solutions encompass previous solutions for one or two‐region flow to a partially penetrating well in a (non‐)leaky confined aquifer. The drawdown response in the abstraction well, non‐Darcy region, and Darcy region is investigated, and sensitivity analysis is made to assess the impact of various controlling parameters. The results demonstrate that the intermediate and late‐stage drawdowns in the abstraction well and non‐Darcy region for the two‐region flow model are larger than the corresponding values for the non‐Darcy flow model. The drawdown in the Darcy region for the two‐region flow model is larger than that for the Darcy flow model throughout the pumping duration. The results of sensitivity analysis show that the drawdowns in the abstraction well, non‐Darcy region, and Darcy region are most sensitive to the non‐Darcy constant n, and is very sensitive to the well configuration, and horizontal hydraulic conductivity of the Darcy region.

Fractional order analysis of unsteady pressure-driven flow in an annulus with momentum slip

Nearly all deformed materials possess both elastic and viscous properties through concurrent storage and dissipation of mechanical vitality. This type of novel functional fluid undergoing deformation is crucial in describing hydrodynamically the general behavior of the fluid. Thus, the working fluid carries memory attribute stored in the particles. The fractional order model for circumferentially pressure-driven flow of viscous fluid in an annulus with momentum slip is investigated theoretically. In particular, an exponentially decaying/growing pressure is considered. For the purpose of this study, the Atangana–Baleanu fractional order model is employed. A solution approach to the fractionalized model is by utilizing Laplace transformation and Tzou’s algorithm for numerical Laplace inversion. It was found that a decisive factor in boosting the speed of flow is suppressing the memory effect of the fluid particle. Also, the instability of the flow can be controlled by enhancing the memory effect.

Consolidation settlement of vertically loaded pile groups in multilayered poroelastic soils

Pile groups are commonly used as the foundations of many structures including those used in transportation infrastructures. Consolidation settlement of a pile foundation is an important design parameter. A theoretical model is developed in this study to estimate the consolidation settlement and axial load transfer of vertically loaded pile groups in multilayered poroelastic soils. The multilayered saturated soil is modeled according to Biot’s poroelasticity theory. In order to determine quasi-static response of pile groups, the interaction problem is first formulated in the Laplace transform domain. Vertical displacement compatibility is enforced at the pile-soil interface to simulate the pile group-soil interaction. Axial deformation of each pile is represented by an exponential series with undetermined coefficients, which are obtained from a variational approach. Vertical displacement influence functions due to a buried uniform vertical load applied to the layered soil are required in the formulation. The application of an exact stiffness matrix method yields the required influence functions. Time-domain solutions are obtained by employing a numerical Laplace inversion method. Numerical results for time-dependent vertical stiffness and consolidation settlement are presented for different pile group configurations, layer profiles, pile elastic moduli and pile lengths.

On the Bimodal Radial Solute Transport in Dual‐Permeability Porous Media

AbstractPorous media usually exhibit distinct transport properties of a dual domain, which is attributed to layering, fracturing, or aggregation. In this study, a mobile‐mobile (MM) model of radial solute transport consisting of two advection‐dispersion equations (ADEs) is presented to describe the bimodal (or double‐peaked) breakthrough curves (BTCs) through two distinct pore domains consisting of interrelated fast‐flow and slow‐flow domains. Transport processes, including advection, dispersion, and first‐order mass transfer between the two domains, were considered in the model. Semi‐analytical solutions were derived using a Laplace transform, matrix diagonalization and numerical Laplace inversion method, and the solutions were verified against the existing analytical solution for the mobile‐immobile (MIM) model under the special case of zero velocity in the slow‐flow domain. In addition, Markov Chain Monte Carlo is applied to estimate the MM model parameters using the experimental data. The results indicate that a larger transfer coefficient between the fast‐flow and the slow‐flow domains resulted in a weaker bimodal phenomenon for concentration distribution curves, which was notably more apparent when there was a greater variation in pore flow velocities between the two domains. Also, the MM model can be used to investigate other types of abnormal transport features such as late arrival and early arrival. Finally, the ADE, MIM, and MM models were used to interpret the experimental data conducted by Fernàndez‐Garcia et al. (2004, http://doi.org.10.1029/2004wr003112), and it was found that the proposed MM model can explain the bimodal feature of BTCs while the ADE and MIM models fail to do so.

Dynamic response of Maxwell fluid in an elastic cylindrical tube

In the present study, the dynamic response of Maxwell fluid in an elastic cylindrical tube is considered. Focusing on the viscoelastic flow through a thin-walled slender elastic cylindrical shell and neglecting inertia in the liquid and solid, a non-homogeneous linear diffusion equation controlling the coupled viscous–elastic system is obtained. The fluid pressure and deformation fields are obtained by numerical Laplace inversion. The results show that the relaxation time of Maxwell fluid has a significant effect on the flow and deformation fields of the viscoelastic system. This research can be used for the design and control of complex time-varying deformation fields and has a certain value for the applications of soft actuators, micro-autonomous systems, and soft robotics.

A mathematical framework for multiphase poromechanics in multiple porosity media

Unconventional geomaterials often exhibit multi-modal pore size distribution. We have developed a comprehensive framework for porous media exhibiting multiple porosity scales that are saturated with one or two types of fluids using mixture theory. Both the governing equations and constitutive laws have been clearly derived and identified, respectively. The effective stress σ′ emerged from the energy balance equation is adoptable for both elastic and elastoplastic deformations, in which pore fractions and saturations play a central role. The proposed model is general in a sense that it works for both uncoupled simulation and coupled simulation. The field equations for uncoupled flow simulation are solved using the Laplace transform and numerical Laplace inversion methods. By visualizing the dimensionless results, we can gain a quantitative insight of the different stages in the depletion process of a naturally fractured reservoir. For coupled flow and geomechanics simulation, a strip load problem and a two-phase flow in a deformable 3D reservoir problem illustrate the impacts of plasticity, multiple porosity, inter-porosity exchange, and capillary pressure on the system response.

An analytical solution for contaminant extraction from multilayered soil using PVD-enhanced system

Prefabricated vertical drains (PVDs) have been employed to enhance the in situ remediation of contaminated fine-grained soils. Subsurface heterogeneity can interfere with the distribution and extraction of contaminants during the remediation process. In the present study, an analytical solution for contaminant extraction from multilayered soil using a PVD-enhanced system is developed based on an equivalent planar two-dimensional model. The analytical solution is derived using a procedure that combines the Laplace transform, eigenfunction method and numerical Laplace inversion. The validity and accuracy of the solution are verified by comparison with an existing analytical solution and the results obtained using a numerical model. The effects of several key parameters on the performance of a PVD remediation system in a triple-layered contaminated soil are evaluated. The results indicate that the remediation efficiency for clay layers decreases with increasing hydraulic conductivity or thickness of the sand layer. The PVD remediation system may be unfeasible for contaminated sites with high subsurface heterogeneity caused by permeability contrast. The remediation process for clay layers can be accelerated by increasing the clay vertical dispersivity.

Dynamic response of cracked non‐homogeneous magneto‐electro‐elastic layer sandwiched by two dissimilar orthotropic layers

AbstractThis paper is concerned with the transient response of cracked functionally graded magneto‐electro‐elastic layer (FGMEE) bonded between dissimilar orthotropic layers under impacts. Fourier and Laplace transforms are employed to obtain the analytical solution of a dynamic magneto‐electro‐elastic dislocation with time‐dependent Burgers' vector at the FGMEE layer. Expressions for stress, electric displacement, and magnetic inductions in the vicinity of the magneto‐electro‐elastic dislocation are derived. The solutions are used to derive singular integral equations for the FGMEE layer containing multiple cracks. Then, the integral equations are solved numerically for the density of dislocations on a crack surface by converting them to a system of linear algebraic equations. Finally, a numerical Laplace inversion algorithm is employed to determine the dynamic stress intensity factor. Numerical examples demonstrate the effects of the non‐homogeneity parameter of the FGMEE layer, applied electric and magnetic loads, and the crack geometry on the dynamic stress intensity. This solution can be used as a Green's function to solve transient problems involving multiple cracks in a functionally graded magneto‐electro‐elastic layer.

Transient behavior of an orthotropic layer with imperfect FGM coating containing multiple interfacial and embedded cracks under anti-plane shear impact load

This paper investigates the transient response of an orthotropic strip with functionally graded (FG) coating weakened by multiple embedded and interfacial cracks. The first part of this paper is concerned with the dynamic response of embedded cracks in the orthotropic substrate with imperfect FG coating. In the second part, the solution of interfacial crack is obtained analytically to extract the dynamic stress intensity factors (DSIFs). In the third part, the proposed method is developed to examine the dynamic fracture behavior of coating–substrate system with an orthotropic FGM coating bonded to an orthotropic substrate through a layer of a bond-coat. In all parts, Laplace and Fourier transforms are used to reduce the problems to a set of singular integral equations with Cauchy kernels which are solved numerically in the Laplace transform domain. The dynamic stress intensity factors are obtained by the numerical Laplace inversion technique. Several examples are solved to obtain DSIFs with various crack lengths and the spring constant of imperfect bonding and material properties of the layers.