iMechanica - Collocation method
https://imechanica.org/taxonomy/term/742
enA Simple Local Variational Iteration Method and Related Algorithm for Nonlinear Science and Engineering
https://imechanica.org/node/23274
<div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/12498">Local variational iteration method</a></div><div class="field-item odd"><a href="/taxonomy/term/12499">Chebyshev polynomial</a></div><div class="field-item even"><a href="/taxonomy/term/742">Collocation method</a></div><div class="field-item odd"><a href="/taxonomy/term/12500">nonlinear differential equation</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><span>A very simple and efficient local variational iteration method for solving problems of nonlinear science is proposed in this paper. The analytical iteration formula of this method is derived first using a general form of first order nonlinear differential equations, followed by straightforward discretization using Chebyshev polynomials and collocation method. The resulting numerical algorithm is very concise and easy to use, only involving highly sparse matrix operations of addition and multiplication, and no inversion of the Jacobian in nonlinear problems. Apart from the simple yet efficient iteration formula, another extraordinary feature of LVIM is that in each local domain, all the collocation nodes participate in the calculation simultaneously, thus each local domain can be regarded as one “node” in calculation through GPU acceleration and parallel processing. For illustration, the proposed algorithm of LVIM is applied to various nonlinear problems including Blasius equations in fluid mechanics, buckled bar equations in solid mechanics, the Chandrasekhar equation in astrophysics, the low-Earth-orbit equation in orbital mechanics, etc. Using the built-in highly optimized <em>ode45</em> function of MATLAB as a comparison, it is found that the LVIM is not only very accurate, but also much faster by an order of magnitude than <em>ode45</em> in all the numerical examples, especially when the nonlinear terms are very complicated and difficult to evaluate.</span></p>
<p><span><a href="https://www.depts.ttu.edu/coe/CARES/pdf/Revisit_LVIM.pdf">https://www.depts.ttu.edu/coe/CARES/pdf/Revisit_LVIM.pdf</a></span></p>
</div></div></div>Fri, 19 Apr 2019 16:18:42 +0000Xuechuan Wang23274 at https://imechanica.orghttps://imechanica.org/node/23274#commentshttps://imechanica.org/crss/node/23274Bifurcation & Chaos in Nonlinear Structural Dynamics: Novel & Highly Efficient Optimal-Feedback Accelerated Picard Iteration Algorithms
https://imechanica.org/node/22379
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/76">research</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/12067">Variational method</a></div><div class="field-item odd"><a href="/taxonomy/term/12068">Picard iteration method</a></div><div class="field-item even"><a href="/taxonomy/term/742">Collocation method</a></div><div class="field-item odd"><a href="/taxonomy/term/12069">structural vibrations</a></div><div class="field-item even"><a href="/taxonomy/term/2453">nonlinear dynamics</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>A new class of algorithms for solving nonlinear structural dynamical problems are derived in the present paper, as being based on optimal-feedback-accelerated Picard iteration, wherein the solution vectors for the displacements and velocities at any time in a finitely large time interval are corrected by a weighted (with a matrix) integral of the error. We present 3 approximations to solve the Euler-Lagrange equations for the optimal weighting functions; thus we present 3 algorithms denoted as Optimal-Feedback-Accelerated Picard Iteration (OFAPI) algorithms-1, 2, 3. The interval in the 3 OFAPI algorithms can be several hundred times larger than the increment required in the finite difference based implicit or explicit methods, for the same stability and accuracy. Moreover, the OFAPI algorithms-2, 3 do not require the inversion of the tangent stiffness matrix, as is required in finite difference based implicit methods. It is found that OFAPI algorithms-1, 2, 3 (especially OFAPI algorithm-2) are far more superior to the currently popular implicit and explicit finite difference methods in terms of computational speed, accuracy, and convergence.</p>
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<p>This paper is online published on "Communications in Nonlinear Science and Numerical Simulation". It can be viewed via the following URL:</p>
<p><span><a href="http://www.depts.ttu.edu/coe/CARES/pdf/Xuechuan_Wang_2018.pdf"><span><strong>http://www.depts.ttu.edu/coe/CARES/pdf/Xuechuan_Wang_2018.pdf</strong></span></a></span></p>
</div></div></div>Thu, 24 May 2018 19:24:41 +0000Xuechuan Wang22379 at https://imechanica.orghttps://imechanica.org/node/22379#commentshttps://imechanica.org/crss/node/22379A novel class of highly efficient and accurate time-integrators in nonlinear computational mechanics
https://imechanica.org/node/20838
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/76">research</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/11503">Strongly nonlinear systems</a></div><div class="field-item odd"><a href="/taxonomy/term/11504">variational iteration method</a></div><div class="field-item even"><a href="/taxonomy/term/742">Collocation method</a></div><div class="field-item odd"><a href="/taxonomy/term/11505">Chebyshev Polynomials</a></div><div class="field-item even"><a href="/taxonomy/term/6745">chaos</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>A new class of time-integrators is presented for strongly nonlinear dynamical systems. These algorithms are far superior to the currently common time integrators in computational efficiency and accuracy. These three algorithms are based on a local variational iteration method applied over a finite interval of time. By using Chebyshev polynomials as trial functions and Dirac–Delta functions as the test functions over the finite time interval, the three algorithms are developed into three different discrete time-integrators through the collocation method. These time integrators are labeled as Chebyshev local iterative collocation methods. Through examples of the forced Duffing oscillator, the Lorenz system, and the multiple coupled Duffing equations (which arise as semi-discrete equations for beams, plates and shells undergoing large deformations), it is shown that the new algorithms are far superior to the 4th order Runge–Kutta and ODE45 of MATLAB, in predicting the chaotic responses of strongly nonlinear dynamical systems.</p>
<p>This paper can be downloaded at: <a href="http://link.springer.com/article/10.1007/s00466-017-1377-4">http://link.springer.com/article/10.1007/s00466-017-1377-4</a></p>
</div></div></div>Tue, 31 Jan 2017 16:40:36 +0000Xuechuan Wang20838 at https://imechanica.orghttps://imechanica.org/node/20838#commentshttps://imechanica.org/crss/node/20838