Due to the uncertainties from modeling, manufacturing, and working environments, many vibration active control systems usually show dynamic uncertain properties. Hence structural reliability ...estimation benchmarking to full-cycle vibratory responses is vitally important. In this study, a novel two-stage dimension-reduced dynamic reliability evaluation (TD-DRE) method for linear quadratic regulator (LQR) controlled structures is developed. This method combines interval uncertainties and the time-variant reliability (TVR) concept. In the first stage, the Taylor series expansion is employed to analyze several typical limit states for definition of the time-discretized dynamic reliability. Then the interval collocation method tackles the solution. In the second stage, the TVR problem is indeed transformed to a time-invariant reliability (TIR) problem. Furthermore, the narrow bounds theorem deduces the presented TD-DRE index. Eventually, two application examples are utilized to verify the effectiveness and accuracy of the proposed method. The proposed TD-DRE is more accurate than the traditional first-order Taylor expansion and more effective than the first-passage reliability evaluation method. This method can provide a reference and an initial value for further design, and improve the efficiency of LQR controller design in practical engineering.
•A novel two-stage dimension-reduced dynamic reliability evaluation method for controlled structure is proposed.•The second-order Taylor expansion and the collocation method are used to calculate the time-discretized dynamic reliability.•The time-variant problem is transformed to a time-invariant problem and the narrow bounds theorem is used to deduce the proposed reliability index.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
A theoretical framework for the accuracy analysis of meshfree collocation methods is presented with a particular focus on the effects of boundary conditions. In the proposed framework, the error of a ...meshfree collocation formulation is decomposed into three parts, namely, the collocation errors associated with the interior nodes, the Dirichlet boundary nodes and the Neumann boundary nodes, respectively. The errors arising from different parts of the computational domain can be conveniently obtained by performing a local truncation error analysis on the discrete meshfree collocation equations corresponding to the generic interior and boundary nodes. It turns out that the accuracy of a meshfree collocation formulation is controlled by the lowest accuracy order resulting from the collocation errors at both interior and boundary nodes, as well as the meshfree interpolation errors. Along this path, it is shown that accuracy of the standard node-based meshfree collocation method is governed by the collocation error related to interior nodes for both L2 and H1 error norms. Nonetheless, for different boundary conditions, the accuracy of the subdomain meshfree collocation method may rely on either the collocation error associated with interior nodes, the collocation errors corresponding to Dirichlet and Neumann boundary nodes, or the interpolation errors. It is further proved that the optimal convergence rates of Galerkin meshfree formulation can be achieved by the subdomain meshfree collocation method if the problems are purely subjected to Dirichlet boundary conditions. The proposed theoretical results are well confirmed by numerical examples.
•A general accuracy analysis framework is proposed for meshfree collocation methods.•This framework particularly takes into account the influence of boundary conditions.•Interior, boundary, and interpolation errors affect the overall collocation accuracy.•Node-based collocation accuracy is governed by interior node error.•Subdomain collocation accuracy relies on either boundary or interpolation errors.
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In this study, the approximations of the higher order linear Fredholm integro-differential-difference equations (IDDEs), with the mixed conditions, have been performed by a new collocation technique ...based on the balancing polynomials. In particular, an attempt has been made to transform the linear IDDEs and the given boundary conditions into matrix equations which corresponds to a system of linear algebraic equations via the proposed procedure. In addition to that, the solutions obtained by the proposed numerical methodology have been analogized with the exact solutions and the error has been registered to manifest the accuracy of the solutions. Furthermore, the reliability and effectiveness of the proposed scheme have been illustrated by some numerical experiments. In addition to that, the error analysis of the technique has been performed along with the investigation of the error function for the improved approximate solutions for the IDDEs. In particular, the rate of approximation of the balancing polynomials has been deduced and the numerical accuracy of the suggested technique has been demonstrated. Apart from that, the absolute errors have been tabulated and graphical figures have been depicted for the solutions obtained via the proposed technique.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
A general class of multi-term nonlinear fractional initial value problems involving variable-order fractional derivatives are considered. Some sufficient conditions for the existence and uniqueness ...of exact solution are established. An hp-version spectral collocation method is presented for solving the problem in numerical frames. It employs the shifted Legendre-Gauss interpolations to conquer the influence of the nonlinear term and the variable-order derivatives. The most remarkable feature of the method is its capability to achieve higher accuracy by refining the mesh and/or increasing the degree of the polynomial. The rigorous error estimates are derived for the problem with smooth solutions on arbitrary meshes and weakly singular solutions on quasi-uniform meshes. Numerical results are given to support the theoretical conclusions.
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BFBNIB, DOBA, GIS, IJS, IZUM, KILJ, KISLJ, NUK, PILJ, PNG, SAZU, UILJ, UKNU, UL, UM, UPUK
In this paper we give an overview of the Method of Fundamental Solutions (MFS) as a heuristic numerical method. It is truly meshless. Its concept and numerical implementation are simple. It has the ...flexibility of using various forms of fundamental solutions, singular, hypersingular, or nonsingular, and mixing with general solutions and particular solutions, for different purposes. The collocation matrix, however, is not guaranteed to be invertible. There are other issues. For example, in using the logarithmic fundamental solution, a degenerate scale can exist, causing the nonuniqueness of solution. For metaharmonic operators, such as the Helmholtz equation, the zeros of the fundamental solutions can create spurious resonance frequencies. These and other issues are discussed, and remedies are offered.
The traditional error analysis for the MFS shows that when the boundary condition is prescribed by a harmonic function, the convergence is exponential either by increasing the number of terms in the approximation, or by increasing the radius of the fictitious boundary. In practical problems, however, a harmonic boundary condition hardly exists. Numerical experiments show that the sources need to stay close to the boundary, and there is an optimal distance. Based on the maximum principle, a posteriori error can be monitored on the boundary to seek the optimal fictitious boundary location. Other topics discussed include the origin of the MFS, the equivalence between the MFS and the Trefftz collocation method, effective condition number, nonsingular MFS, and solving ill-posed and inverse problems.
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This investigation is an attempt to obtain a highly accurate approximation of the spectrum of Sturm–Liouville problems in
ℝ+ by representing the unknown solution of the model in the interpolating ...wavelet basis of
L2(ℝ). To accomplish the goal, the domain
ℝ+ has been stretched to
ℝ to avoid the additional care of the elements in the basis containing boundary point 0. In addition, such transformation may judiciously be utilized to eliminate (up to quadratic) the singularity of the equation. The equation in the new variable has been subsequently transformed into a generalized matrix eigenvalue problem by approximating the new (unknown) function in an appropriate (truncated) basis comprising interpolating scale functions generated by scale functions in Daubechies family. The (interpolating wavelet‐collocation) scheme developed here has been applied to some solvable and quasi‐exactly solvable Sturm–Liouville problems in
ℝ+ appearing in quantum mechanical modeling in flat and curved spaces. It is observed that the approximation of eigenfunctions in the (compact support) interpolating wavelet basis obtained by using the collocation method can be reliably used to reveal a hidden spectrum of quasi‐exactly solvable Sturm–Liouville problems in
ℝ+ with high accuracy.
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FZAB, GIS, IJS, KILJ, NLZOH, NUK, OILJ, SAZU, SBCE, SBMB, UL, UM, UPUK
This study focuses on new numerical approach to the solutions of nonlinear boundary value problems occurring in heat and mass transfer, constructing a matrix-combinatorial method collocated by the ...Chebyshev-Lobatto points and based on the Mittag-Leffler polynomial. For the first time, a matrix-collocation method is coupled with a combinatoric polynomial. In view of this combination, the method converts the linear and nonlinear terms to the matrix forms and then gathers them to a fundamental matrix equation. In addition to the novelty, an inventive nonlinear residual error analysis of general type is firstly theorized and adapted for improving the solutions to the problems in question and also, it allows to regard the nonlinear terms as an operator in calculations. The obtained solutions are thereby corrected. Numerical and graphical illustrations are provided to scrutinize the accuracy, productivity and comparability of the method. Upon evaluations of all these tasks, one can admit that the method is comprehensible, consistent and easily programmable.
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This paper is concerned with the cavity scattering problem in an infinite thin plate, where the out-of-plane displacement is governed by the two-dimensional biharmonic wave equation. Based on an ...operator splitting, the scattering problem is recast into a coupled boundary value problem for the Helmholtz and modified Helmholtz equations. A novel boundary integral formulation is proposed for the coupled problem. By introducing an appropriate regularizer, the well-posedness is established for the system of boundary integral equations. Moreover, the convergence analysis is carried out for the semi- and full-discrete schemes of the boundary integral system by using the collocation method. Numerical results show that the proposed method is highly accurate for both smooth and nonsmooth examples.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
We introduce a class of orthogonal functions associated with integral and fractional differential equations with a logarithmic kernel. These functions are generated by applying a log transformation ...to Jacobi polynomials. We construct interpolation and projection error estimates using weighted pseudo-derivatives tailored to the involved mapping. Then, using the nodes of the newly introduced logarithmic Jacobi functions, we develop an efficient spectral logarithmic Jacobi collocation method for the integrated form of the Caputo–Hadamard fractional nonlinear differential equations. To demonstrate the proposed approach's spectral accuracy, an error estimate is derived, which is then confirmed by numerical results.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
This paper deals with the numerical solution of the general Emden-Fowler equation using the Haar wavelet collocation method. This method transforms the differential equation into a system of ...nonlinear equations. These equations are further solved by Newton's method to obtain the Haar coefficients, and finally the solution to the problem is acquired using these coefficients. We have taken many examples of fifth- and sixth-order equations and implemented our method on those examples. The graphs show the efficiency of the solution for resolution L = 3 and the maximum absolute error of our approach. The error tables give a good picture of the accuracy of this approach.
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BFBNIB, GIS, IJS, KISLJ, NUK, PNG, UL, UM, UPUK
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