Inverse Laplace Transform Research Articles

A hybrid collocation method for the approximation of 2D time fractional diffusion-wave equation

<p>The multi-term time-fractional order diffusion-wave equation (MT-TFDWE) is an important mathematical model for processes exhibiting anomalous diffusion and wave propagation with memory effects. This article develops a robust numerical technique based on the Chebyshev collocation method (CCM) coupled with the Laplace transform (LT) to solve the time-fractional diffusion-wave equation. The CCM is utilized to discretize the spatial domain, which ensures remarkable accuracy and excellent efficiency in capturing the variations of spatial solutions. The LT is used to handle the time-fractional derivative, which converts the problem into an algebraic equation in a simple form. However, while using the LT, the main difficulty arises in calculating its inverse. In many situations, the analytical inversion of LT becomes a cumbersome job. Therefore, the numerical techniques are then used to obtain the time domain solution from the frequency domain solution. Various numerical inverse Laplace transform methods (NILTMs) have been developed by the researchers. In this work, we use the contour integration method (CIM), which is capable of handling complex inversion tasks efficiently. This hybrid technique provides a powerful tool for the numerical solution of the time-fractional diffusion-wave equation. The accuracy and efficiency of the proposed technique are validated through four test problems.</p>

Revisiting Classical Controller Design and Tuning with Genetic Programming.

This paper introduces the application of a genetic programming (GP)-based method for the automated design and tuning of process controllers, representing a noteworthy advancement in artificial intelligence (AI) within the realm of control engineering. In contrast to already existing work, our GP-based approach operates exclusively in the time domain, incorporating differential operations such as derivatives and integrals without necessitating intermediate inverse Laplace transformations. This unique feature not only simplifies the design process but also ensures the practical implementability of the generated controllers within physical systems. Notably, the GP's functional set extends beyond basic arithmetic operators to include a rich repertoire of mathematical operations, encompassing trigonometric, exponential, and logarithmic functions. This broad set of operations enhances the flexibility and adaptability of the GP-based approach in controller design. To rigorously assess the efficacy of our GP-based approach, we conducted an extensive series of tests to determine its limits and capabilities. In summary, our research establishes the GP-based approach as a promising solution for automating the controller design process, offering a transformative tool to address a spectrum of control problems across various engineering applications.

Numerical modelling of advection diffusion equation using Chebyshev spectral collocation method and Laplace transform

In this article a numerical method for numerical modelling of advection diffusion equation is developed. The proposed method is based on Laplace transform (LT) and Chebyshev spectral collocation method (CSCM). The LT is used for time-discretization and the CSCM is used for discretization of spatial derivatives. The LT is used to transform the time variable and avoid the finite difference time stepping method. In time stepping technique the accuracy is achieved for very small time step which results in a very high computational time. The spatial operators are discretized using CSCM to achieve high accuracy as compared to other methods. The method is composed of three primary stages: firstly the given problem is transformed into a corresponding inhomogeneous elliptic problem by using the LT; secondly the CSCM used to solve the transformed problem in LT domain; finally the solution obtained in LT domain is converted to time domain via numerical inverse LT. The inversion of LT is generally an ill-posed problem and due to this reason various numerical inversion methods have been developed. In this article we have utilized the contour integration method which is one of the most efficient methods. The most important feature of this approach is that it handles the time derivative with the Laplace transform rather than the finite difference time stepping approach, avoiding the untoward impact of time steps on stability and accuracy of the method. Five test problems are used to validate the efficiency and accuracy of the proposed numerical scheme.

A 3D time-domain solution for the bending of viscoelastic shells

This article presents a time domain analysis of viscoelastic doubly curved shallow shells employing 3D solution based on equilibrium equations. The constitutive equation is transformed into Laplace domain in order to avoid the time domain integration. The partial differential equations are solved for the mid-surface variables by applying the Navier Technique. The thickness equations are solved using the differential quadrature method (DQM). Lagrange interpolation polynomial are employed as basis functions. Each layer of the panel is discretized by the Chebyshev–Gauss–Lobatto grid distribution. The time-domain displacements are obtained by using the inverse Laplace transformation in an approximate numerical manner. The 3D solution is compared with references in the literature. Given the inclusion of comprehensive 3D time domain solutions, these results can be considered as a benchmark for reference.

Semi‐analytical solutions for the consolidation of unsaturated composite ground with floating permeable columns under equal strain condition

AbstractThis study focuses on the mathematical interpretation and consolidation behavior of unsaturated composite ground with floating permeable columns (FPC). By combining the axisymmetric consolidation model with the equal strain assumption, the consolidation problem of unsaturated composite ground with FPC is transformed into a one‐dimensional consolidation problem of an equivalent double‐layer ground. The consolidation governing equations of the reinforced zone and the underlying stratum are derived respectively. The Laplace transformation and Gauss elimination method are used to obtain solutions for the average excess pore pressure and settlement in the Laplace transform domain. The semi‐analytical solutions for the consolidation of composite ground in the time domain are obtained by implementing the Crump numerical method for the inverse Laplace transformation. A comparison with the existing literature and numerical simulation results from Finite Element Methods (FEM) confirms the correctness of the proposed semi‐analytical solutions. Based on these solutions, the consolidation behavior of an unsaturated composite ground with FPC under instantaneous loading is investigated. The results indicate that the penetration ratio of the permeable column is the most significant factor affecting the consolidation of composite ground. Furthermore, the consolidation rate and settlement of the composite ground also highly depend on the unsaturated soil in the underlying stratum.

An Analytic RLC Model for Coupled Interconnects Which Uses a Numerical Inverse Laplace Transform

An Analytic <i>RLC</i> Model for Coupled Interconnects Which Uses a Numerical Inverse Laplace Transform

A semi-analytical method of three-dimensional dual-phase-lagging heat conduction model

With the development of picosecond laser technology, the dual-phase-lagging (DPL) model is more suitable to describe the heat transfer problem with micro-scale effect and micro-time effect. In this paper, a three-dimensional dual-phase-lagging heat transfer model for picosecond laser processing is established. Then Fourier transform (FT) and Laplace transform (LT) are used to derive the frequency response functions (FRFs) of the problem in the frequency domain, and the inverse Laplace transform (ILT) and discrete-convolution fast Fourier transform (DC-FFT) algorithm are used to solve the temperature field. The calculation accuracy of this method is verified and the effect of phase lag constants on temperature field is studied. The results show that the semi-analytical method has a high accuracy for solving the dual-phase-lagging heat conduction model, and the phase lag of heat flux will affect the heating rate in the heating stage, and the phase lag of temperature gradient will delay the formation of temperature field. The effect of phase lag constants on temperature-time history is investigated and the related phenomena are explained by using the two-step heat transfer theory. This semi-analytical method provides a new approach to solve the problem of three-dimensional dual-phase-lagging heat transfer.

Exact solution of a solar energy model using four different kinds of nanofluids: Advanced application of Laplace transform

This paper analyzes the system of partial differential equations (PDEs) describing the absorption of solar energy via a solar collector utilizing nanofluids. Obtaining exact solution of any system of PDEs is a challenge, however, the current system is effectively solved by means of the Laplace transform (LT). To simplify our procedure, some preliminary theoretical aspects are derived for the inverse LT of some complex expressions and then invested to establish the exact solution. Explicit exact forms are obtained for the involved phenomena in terms of the complementary error function. Moreover, the exact solutions at different special cases are evaluated. Four different kinds of nanofluids are considered to extract the numerical results and physically discussed/interpreted.

Laplace-domain BEM for dynamic analysis of size-dependent couple stress elasticity problems

The size-dependent elastodynamic plane problems are described by a consistent couple-stress theory (C-CST) in this paper. The time-domain basic equations are then transformed to frequency domain by the Laplace transform method. The Laplace-domain boundary integral equations (BIEs) together with the fundamental solutions are derived. Then, these BIEs are numerically solved by a collocation method in conjunction with the numerical treatment of singular integrals. The time-dependent solutions are obtained by the Durbin's method for the inverse Laplace-transform. Some typical size-dependent elastodynamic problems are numerically studied by the developed boundary element method (BEM). Good agreements are observed between the present results and the corresponding analytical or numerical solutions, which validates the high accuracy of the present BEM.

Mathematical significance of strain rate and temperature rate on heat conduction in thermoelastic material due to line heat source

The primary purpose of the present article is to investigate the effect of strain rate and temperature rate factors on a homogeneous, unbounded isotropic elastic medium originating due to continuous line heat source. This study is based on the modified Green-Lindsay Model (MGL) theory proposed by Yu et al. (2018). This model eliminates the discontinuous nature of the displacement field reported under the temperature-rate-dependent thermoelasticity theory (GL) established by Green and Lindsay. The present work obtains the analytical solution for the distributions of stress components, temperature and displacement through the potential function approach accompanied by the Laplace transform method. The inverse Laplace transformation is performed by using short-time approximation method to find the approximated analytical solution of the problem in space-time domain. A detailed analysis of solution is discussed for MGL model and compared to the results predicted by other existing generalized thermoelastic models. The effect of strain-rate and temperature-rate-terms is acknowledged explicitly in mathematical formulation and other significant effects are notified. However, this new model predicts the infinite speed of disturbance analogous to classical theory.

One-dimensional rheological consolidation study on two-sided semi permeable boundary and fractional viscoelastic saturated soil under cyclic loading

Considering the viscoelastic of soft clay, the fractional kelvin model is used to describe the constitutive relationship of soft clay. At the same time, considering that both sides of the soft clay layer have certain permeability, the two-sided semi-permeable boundary is used as its boundary condition. Combined with the cyclic load of different waveforms, the one-dimensional consolidation settlement of saturated soft clay is analyzed. Firstly, the consolidation equation is obtained through Laplace transformation, and the semi-analytic result is obtained by Laplace inverse transformation. The accuracy of the solve method and the model’s applicability are checked by the degradation of boundary conditions and the model. Finally, one-dimensional consolidation settlement under cyclic load with different waveforms is compared.

Thermo-electrical influence of graphene nano-strip on viscothermoelastic nanobeam vibration

In this article, Green-Naghdi type-II heat conduction law was used, and a new analysis of viscothermoelastic, homogeneous, and isotropic nanobeams has been developed. The first end of the beam is based on a graphene nanostrip connected to electricity with a low-voltage current. A thermal load has been applied to the nanobeam due to an electrical current causing a thermal effect. Under simply supported boundary conditions with a fixed aspect ratio, the Green-Naghdi theorem of Type-II has been applied. The Laplace transform domain has been used to calculate the solutions of governing equations. Hoing's approximation approach has been used to calculate the inverse Laplace transformation numerically. In this study, different values of electrical voltage and electrical resistance were used for different cases. Nanobeam functions were found to be significantly influenced by electrical voltage and electrical resistance. Thus, vibration and temperature increment could be controlled by varying electrical voltage and electrical resistance applied to the nanobeam.

Solving Differential Equations and Systems of Differential Equations with Inverse Laplace Transform

The inverse Laplace transform enables the solution of ordinary linear differential equations as well as systems of ordinary linear differentials with applications in the physical and engineering sciences. The Laplace transform is essentially an integral transform which is introduced with the help of a suitable generalized integral. The ultimate goal of this work is to introduce the reader to some of the basic ideas and applications for solving initially ordinary differential equations and then systems of ordinary linear differential equations.

A robust Freeman-Hill-inspired pulse protocol for ringdown-free T1 relaxation measurements

A new difference-spectroscopy method is introduced for measuring T1 relaxation times. It is inspired by the earlier work of Freeman and Hill and eliminates the need for recording signal intensities at thermodynamic equilibrium. The new method is termed SIP-R (Split-Inversion Pulse and Recovery) and reduces the number of refinable parameters in the curve fitting process of relaxation-delay-dependent signal intensities by using two instead of the three parameters typically used in the standard inversion-recovery sequence. The SIP-R method preserves the dynamic range of measurement of the standard inversion-recovery method but converts the rise-to-maximum mathematical functionality of the recorded data into a decay-to-zero functionality. The decay-to-zero functionality renders the SIP-R sequence advantageous for inverse Laplace transformation numerical optimizations. The new technique proves to be extremely robust with respect to pulse imperfections, pulse-power changes during the pulse sequence, pulse-width miscalibrations, resonance offsets, and radiofrequency field variations. It also compensates for acoustic ring-down effects and proves reliable for measurements with inhomogeneously broadened signals up to several kilohertz linewidth. 1H NMR experiments with methane gas at pressures up to 50 atm in toroid-cavity pressure vessel probes and in the presence of the methane-to-methanol conversion catalyst Cu-ZnO/Al2O3 are used to show the usefulness of the new method for relaxation time investigations under pressure, at strong radiofrequency field gradients, and in the presence of paramagnetic materials.

Accurate Modeling and Wave Propagation Analysis of Cracked Slender Structural Members by the Spectral Element Method

The analysis of elastic wave propagation in cracked structures is very useful in the crack detection by the ultrasonic guided wave method. This study presents an accurate spectral element modeling method for cracked slender structural members by using refined waveguide models and a more realistic crack model. Firstly, a spatial spectral beam element model is established for uncracked slender structural member based on the Love rod theory, the modified Timoshenko beam theory, and the Saint-Venant’s torsion theory. Then, the complete local additional flexibility matrix for crack in the structural member with rectangular cross section is derived from the theory of elastic fracture mechanics, and a two-node condensed spectral element model considering the stiffness coupling effect caused by the crack is established for cracked slender structural member. The wave response in cracked structures is solved by the numerical inverse Laplace transformation method. A thorough comparison of the wave responses in cracked structural member evaluated by the presented spectral element model and the 3D solid finite element model is given in the numerical example, which verifies the accuracy and high efficiency of the presented method.

Time-Domain Homogenized Finite Eddy Current Analysis Using Recursive Convolution Method

This paper presents a novel method for effectively and accurately performing time-domain homogenized finite element analysis of multi-turn coil windings. In this method, the macroscopic complex permeability is represented by a partial fraction form, and then inverse Laplace transformation is performed using the recursive convolution method. The proposed method is approximately 100 times faster than conventional finite element analysis, with good agreement between the obtained results. Moreover, the results obtained by the proposed method were also in good agreement with those obtained by measurement.

Viscoelastic Response of a Self‐Gravitational Spherical Earth to Shear Dislocation Obtained Using the Fixed‐Talbot Method

AbstractViscoelastic dislocation theory is important to understanding fundamental geodynamics and validating numerical models in the study of earthquake deformation. Available mathematical solutions differ in assumed Earth geometry and formulation of gravity terms, but the main challenge they commonly face is Inverse Laplace Transform (ILT). Limitations in previously used ILT methods tend to degrade the performance and/or accuracy of the solutions. To overcome these limitations, we have derived new Green's functions for a layered spherical Earth based on the Fixed‐Talbot ILT method. The new solution uses a precise density‐gravity relation, includes pre‐stress advection, buoyancy, and self‐gravitation, invokes the bi‐viscous Burgers rheology, and yields deformational and gravitational changes both at the surface and in the interior of the Earth. The Fixed‐Talbot ILT method, which involves a left‐open parabolic integration path in the complex plane, greatly improves computing efficiency and allows convenient use of a large number of Earth layers. Comparison with other existing solutions demonstrates the excellent performance of the new solution. Taking the postseismic deformation of the 2004 Sumatra Mw 9.2 earthquake as an example, we illustrate how to integrate numerically our Green's functions to represent realistic slip distribution over a three‐dimensionally curved fault. The combination of computational efficiency, mathematical completeness in modeling gravitational effects, adaptability to any radial structure of the Earth, and ability to provide results at any depth including the surface enables a powerful tool for forward and inverse modeling of viscoelastic earthquake cycles in a self‐gravitating spherical Earth.

Time-domain analytical solutions to pulsed eddy current model of moving cylindrical conductor

The real-time or high-speed eddy current testing for conductor specimens are essential in industrial production and the analytical methods of eddy current field for moving conductor need to be further researched. In this paper, the time-domain analytical model of relative motion between the cylindrical conductor and probe is solved under pulsed current excitation. The basic equations and boundary conditions of moving conductor are presented and the boundary-value problem is established. Then the complex frequency-domain expression of voltage induced in the pick-up coil can be obtained by adopting the Laplace and Fourier transformation. Later, the induced voltage in time-domain is derived through the inverse Laplace transformation via residue theorem. Furthermore, the diffusion process of pulsed eddy current is studied by solving the eddy current density in conductor region and the effects of moving velocity on eddy current field are analyzed quantitatively. These calculations show that the larger the moving velocity, the closer the center area of eddy-current distribution is to the pick-up coil, as well as the larger the amplitude of induced voltage in pick-up coil, which can be equivalent to the decrease of axial distance between the pick-up coil and drive coil. The theoretical values obtained in this paper are in excellent agreement with the experimental results and can be applied to the pulsed eddy current testing for moving conductor or probe.

Quantitative stray-field T1 relaxometry with the matrix pencil method

The matrix pencil method (MPM) is tested as an approach to quantitatively process multiexponential low-field nuclear magnetic resonance T1 relaxometry data. The data is obtained by measuring T1 saturation recovery curves in the highly inhomogeneous magnetic field of a stray-field sensor. 0.9% brine solutions, doped with different concentrations of a Gd3+ containing contrast agent, serve as test liquids. Relaxation-times as a function of contrast-agent concentration along with the T1 relaxation curves for combinations of multiple different test liquids are measured, and the results from processing using MPM as well as inverse Laplace transformation as a benchmark are compared. The relaxation-time resolution limits of both procedures are probed by gradually reducing the difference between the relaxation-times of two liquids measured simultaneously. The sensitivity to quantify the relative contribution of each component to the magnetization build-up curve is explored by changing their volume ratio. Furthermore, the potential to resolve systems with more than two components is tested. For the systems under test, MPM shows superior performance in separating two or three relaxation components, respectively and effectively quantifying the time constants.

Non-destructive classification of unlabeled cells: Combining an automated benchtop magnetic resonance scanner and artificial intelligence.

In order to treat degenerative diseases, the importance of advanced therapy medicinal products has increased in recent years. The newly developed treatment strategies require a rethinking of the appropriate analytical methods. Current standards are missing the complete and sterile analysis of the product of interest to make the drug manufacturing effort worthwhile. They only consider partial areas of the sample or product while also irreversibly damaging the investigated specimen. Two-dimensional T1 / T2 MR relaxometry meets these requirements and is therefore a promising in-process control during the manufacturing and classification process of cell-based treatments. In this study a tabletop MR scanner was used to perform two-dimensional MR relaxometry. Throughput was increased by developing an automation platform based on a low-cost robotic arm, resulting in the acquisition of a large dataset of cell-based measurements. Two-dimensional inverse Laplace transformation was used for post-processing, followed by data classification performed with support vector machines (SVM) as well as optimized artificial neural networks (ANN). The trained networks were able to distinguish non-differentiated from differentiated MSCs with a prediction accuracy of 85%. To increase versatility, an ANN was trained on 354 independent, biological replicates distributed across ten different cell lines, resulting in a prediction accuracy of up to 98% depending on data composition. The present study provides a proof of principle for the application of T1 / T2 relaxometry as a non-destructive cell classification method. It does not require labeling of cells and can perform whole mount analysis of each sample. Since all measurements can be performed under sterile conditions, it can be used as an in-process control for cellular differentiation. This distinguishes it from other characterization techniques, as most are destructive or require some type of cell labeling. These advantages highlight the technique's potential for preclinical screening of patient-specific cell-based transplants and drugs.