•Refined FFT algorithm is proposed to remove Gibbs and aliasing effects.•Cartesian system of vector functions is used to decouple the equations.•Dual-variable and position method is employed for ...multilayered system.•Effect of load type on the 3D dynamic Green's function is investigated.•All field quantities induced by dynamic load can be accurately calculated.
By introducing the Cartesian system of vector functions, we investigate the responses of a layered transversely isotropic elastic half-space induced by a dynamic load. The load is vertically applied in the layered half-space. It can be either time-harmonic or horizontally moving with a given velocity. It is demonstrated that, in each layer, in terms of the Cartesian vector functions, we need only a 4 × 4 system of equations for the expansion coefficients of the displacement and traction components, instead of a 6 × 6 system as usually presented. Furthermore, utilizing the vector functions, both the time-harmonic and moving-load cases can be solved in the same way. For the layered case, the previously introduced dual-variable and position method is further used to propagate the solutions from one layer to the other, with unconditional stability. The fast Fourier transform (FFT) is applied to obtain accurate results in the space/time domain efficiently by the numerical inverse Fourier transform, properly developed within the framework of the discrete convolution-FFT (DC-FFT) algorithm. The unique formulation and its corresponding sophisticated algorithm are first validated against existing solutions, and then applied to calculate the displacements and stresses in a layered transversely isotropic half-space. Finally, a typical three-layer flexible pavement structure under a moving circular load is analyzed in detail. Numerical results clearly show the significant effects of both material anisotropy and layering. The present formulation is concise and efficient for the calculation of all the displacements and stresses, which can be applied to the involved engineering practice. The results can also serve as benchmarks for future research in this direction.
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We propose a novel method for solving the static response of a conical indenter on a transversely isotropic and layered elastic half-space. The newly developed Fourier-Bessel series (FBS) system of ...vector functions, along with the unconditionally stable dual-variable and position method, is employed to derive the Green's function in the transversely isotropic and layered elastic half-space under a vertical ring load on the surface. To calculate the response at different field points on the surface, we apply discrete love numbers within the FBS vector system. The load densities in the discretized rings within the contact radius of the conical indenter are determined using the integral least-square method, along with a self-adaptive algorithm developed in this study. Finally, the relationship between the indentation depth (vertical displacement) and the applied load is obtained through force balance between the external load and the summed contact traction. The developed scheme is validated using existing exact solutions for the reduced homogeneous half-space case. Selected numerical results clearly demonstrate the effect of anisotropic material and layering on the indentation response. It is observed that, regardless of whether the structure is a stratified half-space or a layered structure with a rigid substrate, the material properties in the top layer have the most significant influence on the indentation behavior. In the case of a layered structure with an underlying elastic half-space, the material properties in the interlayer and bottom layer could also affect the indentation behaviors.
•Fourier-Bessel series (FBS) system of vector function with Love numbers;•Ring-load Green's functions in layered half-space;•Stable dual-variable and position (DVP) layer matrix method;•Integral least-square method with automatic identification of contact radius;•Numerical results as benchmarks and guidance for layered materials.
Mathematical modeling of multilayered piezoelectric (PE) ceramic substantially acquires attention due to its distinctive advantages of fast response time, positioning, optical systems, vibration ...feedback, and sensors, such as deformation and vibration control. As such, fundamental solution of a PE structure is essential. This paper presents three-dimensional (3D) static and dynamic solutions (i.e. Green’s functions) in a multilayered transversally isotropic (TI) PE layered half-space. The uniform vertical mechanical load, vertical electrical displacement, and horizontal mechanical load are applied on the surface of the structure. The novel Fourier-Bessel series (FBS) system of vector functions (which is computationally more powerful and streamlined) and the dual-variable and position (DVP) method are employed to solve the related boundary-value problem. Two systems of first-order ordinary differential equations (i.e. the LM- and N-types) are obtained in terms of the FBS system of vector functions, with these expansion coefficients being the Love numbers. A recursive relation for the expansion coefficients is established by using DVP method that facilitates the combination of two neighboring layers into a new one and minimizes the computational effort to a great extent. The corresponding physical-domain solutions are acquired by applying the appropriate boundary/interface conditions. Several numerical examples pertaining to static and dynamic response are solved, and the efficiency and accuracy of the proposed solutions are validated with the existing results for the reduced cases. The solutions provided could be beneficial to better developments of PE materials, configurations, fabrication, and applications in the future.
A novel and comprehensive method is proposed for calculating the dislocation Love numbers (DLNs), Green's functions (GFs), and the corresponding deformation in a transversely isotropic and layered ...elastic half-space. It is based on the newly introduced Fourier-Bessel series system of vector functions, along with the dual variable and position method. Two important features associated with this new system are: (1) it is much faster than the conventional cylindrical system of vector functions; (2) we can even pre-calculate the DLNs which are only possible in terms of this new system. This is due to the fact that the variables to be solved in the new system are functions of the simple discrete zero points of the Bessel functions, instead of the numerical integration of continuous Bessel functions between the neighboring zero points as in the conventional system. The introduced dual variable and position method is unconditionally stable as compared to the traditional propagator matrix method in dealing with layering. Exact asymptotic expressions of the DLNs for large wavenumber are further derived, which makes the Kummer's transformation applicable in accelerating the convergence of the corresponding GFs. For the reduced case of homogeneous and isotropic half-space, the present solutions amazingly reduce to the existing exact closed-form solutions. These new features are further seamlessly combined for calculating the deformation due to a finite dislocation (or a finite fault in geophysics) in the layered structure, which are demonstrated to be accurate and efficient.
•Fourier-Bessel series system of vector functions is used to decouple the equations.•Dual-variable and position method is employed for multilayered system.•Poroelastic Green's function due to ...vertical ring load is derived.•Indentation problem is solved via the least-square formulation.•Discrete Love numbers are used to reduce the calculation time.
Poroelastic materials are common in nature and have applications in many engineering fields. In this paper, we derive a general solution of indentation over a multilayered half-space consisting of transversely isotropic and poroelastic materials. The rigid disc-shaped indenter is subjected to a vertical force of Heaviside time-variation. The solution is expressed in terms of the recently introduced powerful Fourier-Bessel series (FBS) system of vector functions combined with the unconditionally stable dual-variable and position method for dealing with layering. Since the problem is a mixed boundary-value one, the Green's functions due to a vertical ring-load are first derived which are then utilized in the integral least-square formulation to derive the solution. In terms of the FBS method, the expansion coefficients, which are further called Love numbers, are discrete, and therefore can be pre-calculated and used repeatedly for different field points on the surface. As such, the solution based on the new FBS method is more efficient and accurate than previous integral-transform methods. This new FBS method is particularly attractive when dealing with mixed boundary-value problems where time-variation is further involved. Numerical examples are conducted to validate the accuracy of the proposed solution and to demonstrate the effects of material layering, geometry, and hydraulic boundary conditions on the contact performance of the material system.
Employing a matrix mask, a vector subdivision scheme is a fast iterative averaging algorithm to compute refinable vector functions for wavelet methods in numerical PDEs and to produce smooth curves ...in CAGD. In sharp contrast to the well-studied scalar subdivision schemes, vector subdivision schemes are much less well understood, e.g., Lagrange and (generalized) Hermite subdivision schemes are the only studied vector subdivision schemes in the literature. Because many wavelets used in numerical PDEs are derived from refinable vector functions whose matrix masks are not from Hermite subdivision schemes, it is necessary to introduce and study vector subdivision schemes for any general matrix masks in order to compute wavelets and refinable vector functions efficiently. For a general matrix mask, we show that there is only one meaningful way of defining a vector subdivision scheme. Motivated by vector cascade algorithms and recent study on Hermite subdivision schemes, we shall define a vector subdivision scheme for any arbitrary matrix mask and then we prove that the convergence of the newly defined vector subdivision scheme is equivalent to the convergence of its associated vector cascade algorithm. We also study convergence rates of vector subdivision schemes. The results of this paper not only establish intrinsic links between vector subdivision schemes and vector cascade algorithms but also improve known results on (generalized) Hermite subdivision schemes and lead to new special types of vector subdivision schemes. Several examples are provided to illustrate the results in this paper on various types of vector subdivision schemes with convergence rates.
Orthogonal and biorthogonal (multi)wavelets on the real line have been extensively studied and employed in applications with success. On the other hand, a lot of problems in applications such as ...images and solutions of differential equations are defined on bounded intervals or domains. Therefore, it is important in both theory and application to construct all possible wavelets on intervals with some desired properties from (bi)orthogonal (multi)wavelets on the real line. Then wavelets on rectangular domains such as 0,1d can be obtained through tensor product. Vanishing moments of compactly supported wavelets are the key property for sparse wavelet representations and are closely linked to polynomial reproduction of their underlying refinable (vector) functions. Boundary wavelets with low order vanishing moments often lead to undesired boundary artifacts as well as reduced sparsity and approximation orders near boundaries in applications. Scalar orthogonal wavelets and spline biorthogonal wavelets on the interval 0,1 have been extensively studied in the literature. Though multiwavelets enjoy some desired properties over scalar wavelets such as high vanishing moments and relatively short support, except a few concrete examples, there is currently no systematic method for constructing (bi)orthogonal multiwavelets on bounded intervals. In contrast to current literature on constructing particular wavelets on intervals from special (bi)orthogonal (multi)wavelets, from any arbitrarily given compactly supported (bi)orthogonal multiwavelet on the real line, in this paper we propose two different approaches to construct/derive all possible locally supported (bi)orthogonal (multi)wavelets on 0,∞) or 0,1 with or without prescribed vanishing moments, polynomial reproduction, and/or homogeneous boundary conditions. The first approach generalizes the classical approach from scalar wavelets to multiwavelets, while the second approach is direct without explicitly involving any dual refinable functions and dual multiwavelets. We shall also address wavelets on intervals satisfying general homogeneous boundary conditions. Though constructing orthogonal (multi)wavelets on intervals is much easier than their biorthogonal counterparts, we show that some boundary orthogonal wavelets cannot have any vanishing moments if these orthogonal (multi)wavelets on intervals satisfy the homogeneous Dirichlet boundary condition. In comparison with the classical approach, our proposed direct approach makes the construction of all possible locally supported (multi)wavelets on intervals easy. Seven examples of orthogonal and biorthogonal multiwavelets on the interval 0,1 will be provided to illustrate our construction approaches and proposed algorithms.
The subject of research in the article is sigularly perturbed controllable systems of differential equations containing terms with a small parameters on the right-hand side, which are not completely ...known, but only satisfy some constraints. The aim of the work is to expand the study of the behavior of solutions of singularly perturbed systems of differential equations to the case when the system is influenced not only by dynamic (small factor at the derivative) but also parametric (small factor at the right side of equations) uncertainties and to determine conditions under which such systems will be asymptotically resistant to any perturbations, estimate the upper limit of the small parameter, so that for all values of this parameter less than the obtained estimate, the undisturbed solution of the system was asymptotically stable. The following problems are solved in the article: singularly perturbed systems of differential equations with regular perturbations in the form of terms with a small parameter in the right-hand sides, which are not fully known, are investigated; an estimate is made of the areas of asymptotic stability of the unperturbed solution of such systems, that is, the class of systems that can be investigated for stability is expanded, the formulas obtained that allow one to analyze the asymptotic stability of solutions to systems even under conditions of incomplete information about the perturbations acting on them. The following methods are used: mathematical modeling of complex control systems; vector Lyapunov functions investigation of asymptotic stability of solutions of systems of differential equations. The following results were obtained: an estimate was made for the upper bound of a small parameter for sigularly perturbed systems of differential equations with fully known parametric (fully known) and dynamic uncertainties, such that for all values of this parameter less than the obtained estimate, such an unperturbed solution is asymptotically stable; a theorem is proved in which sufficient conditions for the uniform asymptotic stability of such a system are formulated. Conclusions: the method of vector Lyapunov functions extends to the class of singularly perturbed systems of differential equations with a small factor in the right-hand sides, which are not completely known, but only satisfy certain constraints.
Let A be an n-by-n matrix and M(x,y,z)=zIn+xℜ(A)+yℑ(A), where ℜ(A)=(A+A⁎)/2 and ℑ(A)=(A−A⁎)/(2i). The inverse numerical range problem seeks a unit vector x corresponding to a given point z of the ...numerical range of A satisfying z=x⁎Ax. A kernel vector function ξ=ξ(x,y,z) of M(x,y,z) with point (x,y,z) on the curve FA(x,y,z)=det(M(x,y,z))=0 plays the role of the unit vector x for the inverse numerical range. The columns of the adjugate matrix L(x,y,z)=(Ljk(x,y,z)) of M(x,y,z) are kernel vector functions of M(x,y,z). We prove the Abel theorem on the intersections of the algebraic curves FA(x,y,z)=0 and Ljk(x,y,z)=0. A concrete numerical example is provided to verify the result using the Maple package algcurves.
Time-harmonic loading over layered elastic half-spaces has applications in various science and engineering fields. While various approaches have been proposed in solving the related boundary-value ...problems, in this paper, we propose a new approach, which is based on the novel Fourier-Bessel series system of vector functions and the dual variable and position method (DVP). While the DVP method was proposed recently and verified to be computationally stable and efficient, the Fourier-Bessel series system of vector functions is newly introduced. Similar to the cylindrical system of vector functions, the normal (dilatational) and shear (torsional) deformations (waves) can be separated and solved in terms of the LM- and N-types of the new vector function system. The new formulation is coded, and the corresponding algorithm/program is applied to a couple of cases. It is shown that, by comparing previous approaches, this new series system of vector functions is equally accurate, but much more computationally powerful. Since it is substantially time saving in calculation, it is hopeful that this new approach would have broad applications related to transient response and inverse problems in elastodynamics of layered systems.
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