Numerical Laplace Inversion Research Articles

Rapid Solution of Stiff Differential Equations and Accurate Numerical Laplace Inversion of Steep and Oscillatory Functions using IMN Approximants

AbstractIMN approximants are a fast and convenient method of solving initial value problems in linear stiff differential algebraic equations as well as obtaining the numerical inversion of Laplace transforms In the past it was impossible to use them to obtain sufficiently accurate inversions of certain steep and highly oscillatory responses as useable values of N had to be relatively small not only to ensure reliable evaluation of IMN constants but also in order to avoid undue rounding errors in the computed results However, the development of computer algebra systems such as Mathematica which permit infinite precision computation has provided greater latitude for the application of the method This work is an exposition of the potency of IMN approximants in accurately and cheaply inverting functions in the Laplace domain whose time functions are steep, oscillatory or stiff Previous applications of the method to some examples in the literature led to wrong conclusions as the capabilities of the method were not fully explored We show how to obtain very accurate results in these circumstances using both the global and step‐by‐step methods The results of using IMN step‐by‐step technique to rapidly solve stiff differential equations of a large staged process are also presented As Matlab gives inaccurate results, Mathematica has been used to compute both the transfer function and the analytical expression for the time response of the plant Extended values ofM and N as well as the ranges of the corresponding IMN constants are tabulated for IMN approximants of full grade.

Dynamic behavior of a cylindrical crack in a functionally graded interlayer under torsional loading

In this paper the problem of a cylindrical crack located in a functionally graded material (FGM) interlayer between two coaxial elastic dissimilar homogeneous cylinders and subjected to a torsional impact loading is considered. The shear modulus and the mass density of the FGM interlayer are assumed to vary continuously between those of the two coaxial cylinders. This mixed boundary value problem is first reduced to a singular integral equation with a Cauchy type kernel in the Laplace domain by applying Laplace and Fourier integral transforms. The singular integral equation is then solved numerically and the dynamic stress intensity factor (DSIF) is also obtained by a numerical Laplace inversion technique. The DSIF is found to rise rapidly to a peak and then reduce and tend to the static value almost without oscillation. The influences of the crack location, the FGM interlayer thickness and the relative magnitudes of the adjoining material properties are examined. It is found among others that, by increasing the FGM gradient, the DSIF can be greatly reduced.

Higher order analysis of the distribution of crystallization processes in metallic glasses

Metallic glassy alloys prepared by rapid quenching are expected to exhibit, unlike conventional polycrystalline matter, several specific features. The existence of disorder (frozen metastable state), chemical SRO, structural SRO (clusters), local heterogeneity, occurrence of ordered, yet noncrystalline, groups of atoms in the amorphous matter necessarily must be reflected in the complexity of processes controlling crystallization. Experimental observations on crystallization of amorphous alloys interpreted via classical thermodynamic analyses (which result usually in a single value, sum of values or prescribed activation energy distribution) provide further indication that these processes are numerous and should be more realistically described by a distribution reflecting the thermodynamic distribution of the initial state of the structure. These processes are by no means sufficiently described by standard simple theoretical distributions. Using a numerical Laplace inversion method, it has been possible to determine the distribution of processes controlling crystallization of metallic glasses in time and parametrically against temperature without assumptions concerning the form of the distributions. The results obtained from the changes of electrical resistivity in the course of isothermal crystallization of Fe–Co–B metallic glass have been analysed with respect to the expected inherent complexity of the system.

Dynamic stress intensity factor of a functionally graded material under antiplane shear loading

The dynamic response of a finite crack in an unbounded Functionally Graded Material (FGM) subjected to an antiplane shear loading is studied in this paper. The variation of the shear modulus of the functionally graded material is modeled by a quadratic increase along the direction perpendicular to the crack surface. The dynamic stress intensity factor is extracted from the asymptotic expansion of the stresses around the crack tip in the Laplace transform plane and obtained in the time domain by a numerical Laplace inversion technique. The influence of graded material property on the dynamic intensity factor is investigated. It is observed that the magnitude of dynamic stress intensity factor for a finite crack in such a functionally graded material is less than in the homogeneous material with a property identical to that of the FGM crack plane.

Ultrafast optical dynamics of spiro-compounds

We report on time-resolved measurements of spiro-type molecules in the ps- and fs-timescale. Due to the orthogonality of the transition dipoles of two identical moieties connected through a spiro-atom, one expects these compounds to show two perpendicular, degenerate states that may enter into a complex free evolution upon fs excitation. On ps scales the bichromophoric units behave as isolated absorbers, thus their fluorescence stems from localized site populations. Level relaxation gives rise to slightly non-exponential relaxation that can be quantified in terms of underlying life-time distributions by employing numerical Laplace inversion. Fs fluorescence upconversion experiments have been employed to elucidate, for the first time, nuclear and electronic dynamics subsequent to broad-band fs-preparation.

Identification of Dynamic Pressure Distribution Applied to the Elastic Thin Plate.

This paper deals with a strategy for identification of dynamic pressure distribution applied to the elastic thin plate. Boundary element method is adopted to obtain discretized matrix relation between input loading and output signal, where strain information is used as a supplementary information of the inverse analysis in the present study. Laplace transform and numerical inverse Laplace inversion are introduced in order to treat the dynamic behavior. The coefficient matrix to be solved is given on the Laplace-transformed domain as the form of transfer function relating input and output signals. Since the inverse analysis requires regularization to stabilize the ill-posed solutions, Tikhonov regularization has been employed with singular value decomposition, where the optimal parameter of Tikhonov method was determined by Hansen's L-curve method. Through some numerical simulation on the circular plate subjected to dynamically distributed loading, the usefulness of the present method based on the Laplace-transformed-BEM is demonstrated in detail.

Impact Response of a Piezoelectric Ceramic Strip with a Central Crack.

The plane strain dynamic singular stress problem for a piezoelectric ceramic plate having a central crack is considered. The Laplace and Fourier transform techniques are used o formulate the problem in terms of a singular integral equation. The singular integral equation is solved by using the Gauss-Jacobi integration formula. A numerical Laplace inversion routine is used to recover the time dependence of the solution. Numerical calculations are carried out, and the main results presented are the variation of the stress intensity factor as functions of the geometric parameters and the piezoelectric material properties of the plate.

Electroelastic fracture dynamics for multilayered piezoelectric materials under dynamic anti-plane shearing

This paper describes a method to analyze the dynamic response of a multilayered piezoelectric material plate containing some non-collinear cracks. It is assumed that the multilayers is composed of numerous laminae with cracks located at the interface of composite layers. Based upon Fourier transforms and Laplace transforms, the boundary value problem is reduced to a system of generalized singularity integral equations in the Laplace transform domain. By utilized numerical Laplace inversion, the time-dependent full field solutions are obtained in the time domain. Numerical results are plotted to illustrate how the loading state and material non-homogeneity influence the stress fields and the electric displacement fields ahead of the crack tip.

Antiplane mechanical and inplane electric time-dependent load applied to two coplanar cracks in piezoelectric ceramic material

This work is concerned with the dynamic response of two coplanar cracks in a piezoelectric ceramic under antiplane mechanical and inplane electric time-dependent load. The cracks are assumed to act either as an insulator or as a conductor. Laplace and Fourier transforms are used to reduce the mixed boundary value problems to Cauchy-type singular integral equations in Laplace transform domain. A numerical Laplace inversion algorithm is used to determine the dynamic stress and electric displacement factors that depend on time and geometry. A normalized equivalent parameter describing the ratio of the equivalent magnitude of electric load to that of mechanical load is introduced in the numerical computation of the dynamic stress intensity factor (DSIF) which has a similar trend as that for the pure elastic material. The results show that the dynamic electric field will impede or enhance crack propagation in a piezoelectric ceramic material at different stages of the dynamic electromechanical load. Moreover, the electromechanical response is greatly affected by the ratio of the crack length to the ligament between the cracks. The stress and electric displacement intensity factor can be combined by the energy density factor or function to address the fracture of piezoelectric materials under the combined influence of electromechanical loading.

Discrimination of defects in III–V semiconductors by positron lifetime distribution

In this paper, the numerical Laplace inversion technique and maximum entropy method are utilized to extract continuous positron lifetime distribution in semiconductors. The result is used to discriminate the native vacancy-type defects in as-grown GaAs and InP with different conduction type. Direct evidence of shallow positron traps were also observed in ion-implanted p-InP. It is demonstrated that the lifetime distribution can give us more detailed information on the native defects.

Concentration and temperature transients in heterogeneous porous media: Part I: Linear transport

The transport of solute/heat in porous media is modeled by the convection–dispersion equation. In the study of such transport processes, while the resident concentration has always been used to develop the governing differential equations, the flux concentration has been the most commonly measured quantity in experiments. Hence, earlier studies have emphasized that care must be exercised in the quantitative interpretation of experiments to avoid inconsistent use of solutions with the actual conditions of experiments. They have further demonstrated that performing an actual variable transformation in the homogeneous medium differential equation of resident concentrations shows that the flux concentration also satisfies the convection dispersion equation including those with linear reaction. Hence, the physical meanings of solutions with respect to these two concentrations are inferred from boundary conditions. The earlier studies have also provided a classification of solutions of homogeneous medium models based on these two concentration variables. This work first generalizes the theory of dependent variable transformation for dispersive transport and further extends it to include the heterogeneous medium models that assume a diffusive transport between the fracture and matrix phases. Then, two new solutions of the heterogeneous medium solutions are derived. Hence, the experimentalist is furnished with the heterogeneous medium model solutions in which at least one of them is consistent with the actual conditions of an experiment. Secondly, in this work, the concept of block geometry functions (BGFs) is extended to include frequency distributions of multiple block sizes more likely to exist in heterogeneous media than a single block. It was found that BGFs of various distributions with λ min less than 0.1 differ only slightly, and hence, may be represented by the BGF of the mean block size. Otherwise care must be exercised to include the block size distribution effect. In addition, numerical Laplace inversion of complete solutions having those slightly differing BGFs is found to lead to significant differences for cases where mobile and immobile phase fractions are close. Finally, interpretation of tracer return profiles in a heterogeneous system by employing a nonlinear regression technique is illustrated. Simulated field data are matched by a four-parameter theoretical model and sensitivity of results to parameter values is investigated.

Elastodynamic Fracture Analysis of Multiple Cracks by Laplace Finite Element Alternating Method

The Laplace finite element alternating method, which combines the Laplace transform technique and the finite element alternating method, is developed to deal with the elastodynamic analysis of a finite plate with multiple cracks. By the Laplace transform technique, the complicated elastodynamic fracture problem is first transformed into an equivalent static fracture problem in the Laplace transform domain and then solved by the finite element alternating method developed. To do this, an analytical solution by Tsai and Ma for an infinite plate with a semi-infinite crack subjected to exponentially distributed loadings on crack surfaces in the Laplace transform domain is adopted. Finally, the real-time response can be computed by a numerical Laplace inversion algorithm. The technique established is applicable to the calculation of dynamic stress intensity factors of a finite plate with arbitrarily distributed edge cracks or symmetrically distributed central cracks. Only a simple finite element mesh with very limited number of regular elements is necessary. Since the solutions are independent of the size of time increment taken, the dynamic stress intensity factors at any specific instant can even be computed by a single time-step instead of step-by-step computations. The interaction among the cracks and finite geometrical boundaries on the dynamic stress intensity factors is also discussed in detail. [S0021-8936(00)02103-6]

Application of positron lifetime distribution to the discrimination of defects in semiconductors

In this paper, two methods of extracting continuous positron lifetime distribution are utilized, one is the numerical Laplace inversion, another is the maximum entropy method. The result of continuous lifetime distribution is used to discriminate the native vacancy-type defects in GaAs and InP with different conduction type. It is demonstrated that the lifetime distribution can give us more detailed information on the native defects. In particular, the direct evidence of positron trapping by vacancies is observed both in semi-insulating GaAs and in p-type InP, of which the identification of defects is still a question in debate.

Cracks problem for non-homogeneous composite material subjected to dynamic loading

In this paper, we present a method to analyse the dynamic and steady response of non-homogeneous composite materials. Differing from the existing works reported in literature, the present method can be used for arbitrarily varying material properties through thickness direction and the crack number can be larger than one. It is assumed that the composite material is orthotropic and all the material properties depend only on the coordinates y (along the thickness direction). The material non-homogeneity is simulated by dividing the plate into a number of layers, each layer is assigned slightly different material properties. The method is based upon the Fourier and Laplace transforms to reduce the boundary value problem to a system of generalized singularity integral equations in the Laplace transform domain. The singular integral equations for the problem are derived and numerically solved by weighted residual value methods. By utilized numerical Laplace inversion the time-dependent full field solutions are obtained. As the numerical illustrates, three different cracked specimens, a functionally graded material, a metal-ceramic joint with functionally graded interlayer, and a metal substrate/functionally graded film structure are presented for various material non-homogeneity parameters and/or functionally graded layer thickness. The results obtained demonstrate that the present model is an efficient tool in the fracture analysis of composite materials with properties varying in the thickness direction.

Impact response of a finite crack in an orthotropic piezoelectric ceramic

The transient dynamic stress intensity factor and dynamic energy release rate were determined for a cracked piezoelectric ceramic under normal impact in this study. A plane step pulse strikes the crack and stress wave diffraction takes place. Laplace and Fourier transforms are employed to reduce the transient problem to the solution of a pair of dual integral equations in the Laplace transform plane. The solution of the dual integral equations is then expressed in terms of a Fredholm integral equation of the second kind. A numerical Laplace inversion technique is used to compute the values of the dynamic stress intensity factor and the dynamic energy release rate for some piezoelectric ceramics, and the results are graphed to display the electroelastic interactions.

Dynamic response of a pile in a multi-layered soil to transient torsional and axial loading

The dynamic response of an elastic pile subjected to transient torsional and axial loading is considered. The pile is embedded in a multi-layered elastic soil. The stress field in a soil is simplified by following the existing solutions for a pile subjected to static torsional and axial loading. The dynamic equilibrium equation of soil is solved in the Laplace domain by using analytical techniques. The soil resistance is coupled into a one-dimensional governing equation of a pile segment and an analytical solution is presented. An impedance matrix can then be derived for a pile segment relating end stress resultants to the displacements. Non-homogeneous initial conditions of the pile are considered. The impedance matrices of pile segments are assembled by following the concepts of finite element method to analyse a pile embedded in a multi-layered soil. The treatment of soil base response is also discussed. Time domain solutions are obtained by using a numerical Laplace inversion procedure. The extension of the analysis to linear viscoelastic soils using the correspondence principle in the theory of viscoelasticity is discussed. Selected numerical solutions for a pile embedded in a homogeneous soil and a &squo;Gibson’ soil are presented to portray the influence of pile moduli and slenderness ratios, loading history and soil non-homogeneity on the dynamic response of an elastic pile. Nous étudions la réponse dynamique d’un pilot élastique soumis à une charge de torsion et axiale transitoire. Le pilot est enfoui dans un sol élastique constittué de plusieurs couches. Nous simplifions le champ de contrainte dans un sol en appliquant les solutions existantes pour un pilot soumis à une charge de torsion et axiale statique. L’équation de 1’équilibre dynamique du sol est résolue dans le domaine de Laplace en utilisant des techniques analytiques. Nous couplons la résistance du sol en une équation standard unidimensionnelle pour un segment de pilot et nous présentons une solution analytique. Une matrice d’impédance pent alors être dérivée pour un segment de pilot en liant les résultantes de contrainte en bout aux déplacements. Nous examinons les conditions initiales non homogènes du pilot. Pour analyser un pilot enfoui dans un sol constitué de plusieurs couches, nous assemblons les matrices d’impédance des segments de pilot en suivant les concepts de la méthode d’éléments finis. Nous discutons aussi du traitement de la réponse de base du sol. Les solutions du domaine de temporisation sont obtenues en utilisant une procédure d’inversion numérique de Laplace. Nous discutons de l’extension de l’analyse aux sols à viscoèlasticit&acutee;aire en utilisant le principe de correspondance dans la théorie de viscoélasdcité. Nous présentons de solutions numériques choisies pour un pilot enfoui dans un sol homogène et un sol &squo;Gibson’ pour décrire l’influence des modules de pilot et des coefficients de gracilité, des comportements aux charges et de la non homégénéité du sol sur la réponse dynamique d’un pilot élastique.

Unimolecular phase space theory rates by inversion of angular momentum-conserved partition function

A simplified phase space theory (PST) of angular momentum-conserving microcanonical rate constant at specified total angular momentum J in unimolecular fragmentation under a central potential is proposed via the reverse association of fragments. Angular momentum-conserved rotational–translational sum/density of states of fragments is approximated by interpolation between "‘high-J'' and ""low-J'' states (Chem. Phys. Lett., 1996, 262, 539), from which is obtained in closed form the corresponding J-conserved partition function Qxi(J); this represents the core result of this work [eqn. (20)]. A relatively simple numerical Laplace inversion routine of the product of Qxi(J) and the vibrational partition function accomplishes in a single stroke the inversion that leads immediately to the microcanonical rate constant k(E,J)PST. Averaging of Qxi(J) over J leads directly to the canonical (thermal) PST rate constant for dissociation. The procedure is checked against available more elaborate PST results and is illustrated on cases representing five different combinations of fragment symmetries: linear+atom, sphere+atom, linear+linear, sphere+linear and sphere+sphere. The method requires minimal computational effort and is particularly efficient for calculations involving large molecules and large angular momenta.

OUT-OF-PLANE DYNAMIC RESPONSES OF NON-CIRCULAR CURVED BEAMS BY NUMERICAL LAPLACE TRANSFORM

The paper presents a systematic method for analyzing the out-of-plane dynamic behaviours of non-circular curved beams the governing equations of which take into account the effects of shear deformation, rotary inertia, and viscous damping. The procedure consists of formulating an analytical solution for the transformed governing equations in the laplace transform domain and computing the responses in the time domain by means of numerical Laplace inversion. As key elements in the dynamic stiffness method, the first known transformed dynamic stiffness matrix and equivalent nodal loading vector for non-circular curved beams subjected to distributed external loading are established from the analytical solution developed using the famous Frobenius method. With a simple modification, the formulation of the solution is also suitable for free vibration analysis, which results in an exact solution. As numerical examples for transient analysis, time responses of displacement components and stress resultants are found for a two-span elliptic beam subjected to a rectangular impulse. The behaviours of the response due to the variation of the ratio of the long axis to the short one are investigated.

Regularization of Numerical Inversion of the Laplace Transform for the Inverse Analysis of Impact Force.

This paper is concerned with the inverse problem of impact force, which is to estimate the impact forces acting on bodies from the measurement of their impact responses. As the authors have shown previously, both numerical Laplace transformation and its inversion are required to solve the inverse problem. Since the inverse analysis is based on experimental data and, in addition, the Laplace inversion is typical of ill-posed problems, straightforward computation tends to provide an unacceptable estimate of the true impact force. Therefore, regularization of the Laplace inversion is necessary to obtain a good estimate of the impact force. In this paper Tikhonov regularization is applied to the numerical Laplace inversion using FFT. Both numerical simulation and experimental results verify that the Tikhonov regularization is effective for improving the accuracy of the estimated impact force. In addition, it is shown that Hansen's L-curve method can be employed to determine appropriately the regularization parameter.

Analysis of general second-order fluid flow in double cylinder rheometer

The fractional calculus approach in the constitutive relationship model of second-order fluid is introduced and the flow characteristics of the viscoelastic fluid in double cylinder rheometer are studied. First, the analytical solution of which the derivative order is 1/2 is derived with the analytical solution and the reliability of Laplace numerical inversion based on Crump algorithm for the problem is verified, then the characteristics of second-order fluid flow in the rheometer by using Crump method is analyzed. The results indicate that the more obvious the viscoelastic properties of fluid are, the more sensitive the dependence of velocity and stress on fractional derivative order is.