iMechanica - Comments for "Constrained Moving Least Squares (CMLS) Method"
http://imechanica.org/node/4977
Comments for "Constrained Moving Least Squares (CMLS) Method"enRe: max-ent in DLSM meshfree
http://imechanica.org/comment/10634#comment-10634
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<p><em>In reply to <a href="http://imechanica.org/node/4977">Constrained Moving Least Squares (CMLS) Method</a></em></p>
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<font size="2">Dear Jafar,</font>
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<font size="2">1) In principle, whenever you can use MLS you can use maxent basis functions, as long as the approximation is linear. As you may notice in the papers by Prof. Arroyo and Prof. Sukumar, only linear approximations are currently developed since, for example, second-order approximation represents an non-feasible solution to the constrained maximization problem. However, I knew that Prof. Arroyo has developed a second-order maxent approximation which will be described in a subsequent paper to appear soon, I guess. </font>
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<font size="2">2) I cannot answer this since I am not familiar with the problem you are trying to solve. If you need to compute second-order derivatives, then you can use the code that Professor Sukumar posted in his blog here in iMechanica. At the moment, I just have used first order derivatives, so I cannot say much about second-order derivatives.</font>
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<font size="2">3) Sure. I am still learning things about maxent, so I see in this an opportunity to learn more about the same.</font>
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<font size="2">Regards,</font>
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<font size="2">Alejandro. </font>
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</ul>Fri, 24 Apr 2009 18:36:51 +0000Alejandro Ortiz-Bernardincomment 10634 at http://imechanica.orgmax-ent in DLSM meshfree
http://imechanica.org/comment/10629#comment-10629
<a id="comment-10629"></a>
<p><em>In reply to <a href="http://imechanica.org/node/4977">Constrained Moving Least Squares (CMLS) Method</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
<span>Dear <span>Alejandro</span></span>
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Excuse me. I was working in my thesis (Adaptive Refinement with mixed formulation) and CMLS approximation since past 5 weeks. My M.Sc. thesis finished. Now I want to use max-ent approach in discrete least square meshless method. I have some questions about max-ent approach?
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1) Can I use max-ent approach instead of MLS for shape functions approximation in any meshfree methods?
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2) Results of meshfree methods in irreducible formulation with MLS approximation for second derivatives of shape functions (stresses) have some error compared with analytical solution. Is this problem exist with max-ent approach?
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3) Can you guide me in this work?
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Best Regard
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Jafar Amani
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</ul>Thu, 23 Apr 2009 21:51:03 +0000Jafarcomment 10629 at http://imechanica.orgCMLS file attached
http://imechanica.org/comment/10178#comment-10178
<a id="comment-10178"></a>
<p><em>In reply to <a href="http://imechanica.org/node/4977">Constrained Moving Least Squares (CMLS) Method</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
<span>Dear <span>Alejandro</span></span>
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CMLS file is attached?
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Thank
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</ul>Thu, 19 Mar 2009 07:24:12 +0000Jafarcomment 10178 at http://imechanica.orgRe: CMLS or max-ent in mixed formulation?
http://imechanica.org/comment/10147#comment-10147
<a id="comment-10147"></a>
<p><em>In reply to <a href="http://imechanica.org/comment/10139#comment-10139">CMLS or max-ent in mixed formulation?</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p><font size="2">Can you attach the file again? ... It seems there is no attachement.</font></p>
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</ul>Wed, 18 Mar 2009 05:40:26 +0000Alejandro Ortiz-Bernardincomment 10147 at http://imechanica.orgCMLS or max-ent in mixed formulation?
http://imechanica.org/comment/10139#comment-10139
<a id="comment-10139"></a>
<p><em>In reply to <a href="http://imechanica.org/comment/10131#comment-10131">Re: Penalty coefficient & max-ent in meshless with mixed formula</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p><span>Dear <span>Alejandro</span></span><span>1)In <span> </span>solving the differential equations governing the planar elasticity problems,<span> </span>order of approximation of stresses is 2, therefore second derivative of MLS shape functions should be used. In paper with entitle: “Discrete least squares meshless method with sampling points for the solution of elliptic partial differential equations“ by Rahmani & Afshar was shown that second derivative of MLS shape functions have more discontinuity than first derivatives. Since in mixed formulation the resulting governing equations are of the first order, both the displacements and stress boundary conditions are of the Dirichlet type. <span> </span>Thus in mixed formulation order of approximation decrease to 1, therefore higher accuracy is obtained. only in the paper by Prof. Atluri “Meshless Local Petrov-Galerkin (MLPG) mixed collocation method for elasticity problems” mixed formulations in meshless methods was used.</span><span> </span><span>2) No, penalty coefficient method is differing from <span>Lagrange multipliers. In Liu’s book “mesh free methods - moving beyond the finite element method” have been explained that in using Lagrange multipliers stiffness matrix may have nonsymmetrical, while in using penalty coefficient method in my approach stiffness matrix is symmetrical. Also the computational time maybe increase when using Lagrange multipliers specially in using sampling points with nodal points.<span> </span></span></span><span> </span><span><span> </span>3)No, </span><span>mixed method variously used in finite element method for analysis of plane elasticity problems. For stability the finite element spaces are required to satisfy the Ladyzhenskaya-Babuska-Brezzi (LBB) condition . But <span>in mixed formulation only formulation was changed compared with previous formulation (Irreducible formulation). Results of mixed formulation are better compared with irreducible formulation when amount of penalty coefficients have been determined with some efforts. I want to Solve the penalty problem with:</span></span><span>1)Change MLS approximation to max-ent approximation to avoid penalty coefficient.</span><span>2) Change MLS approximation to CMLS approximation to avoid penalty coefficient. The formulation of CMLS from Liu’s book is attached.</span><span>Which of the above can solve imposing BC’s in mixed formulation?</span><span>Kindest regard</span><span>Jafar</span><span></span></p>
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</ul>Tue, 17 Mar 2009 10:35:53 +0000Jafarcomment 10139 at http://imechanica.orgRe: Penalty coefficient & max-ent in meshless with mixed formula
http://imechanica.org/comment/10131#comment-10131
<a id="comment-10131"></a>
<p><em>In reply to <a href="http://imechanica.org/node/4977">Constrained Moving Least Squares (CMLS) Method</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
<font face="arial,helvetica,sans-serif" size="2">Dear Jafar,</font>
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<font face="arial,helvetica,sans-serif" size="2">These are my replies to your questions: <br /></font>
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<font face="arial,helvetica,sans-serif" size="2">1) Do you mean number of lagrange multipliers? if so, it should be equal to the number of essential boundary conditions you have. </font>
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<font face="arial,helvetica,sans-serif" size="2">2) Yes, you don't need to enforce essential boundary conditions with lagrange multipliers. Maximum-entropy basis functions vanish at the boundary naturally. That is because they are derived from a convex optimization problem (you may see the proof in the paper by Arroyo & Ortiz, 2006). Just apply them in the same way as in FE, and of course you can replace your MLS basis functions with max-ent basis functions with the advantange that you will not need to enforce essential boundary conditions. Also using MLS you need to invert the moment matrix for every evaluation point. On using max-ent you just need to solve a system of "n" nonlinear equations where "n" is the spatial dimension for every evaluation point. Thus, simpler and more robust. </font>
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<font face="arial,helvetica,sans-serif" size="2">Now, some questions for you: </font>
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<font face="arial,helvetica,sans-serif" size="2">1) What order is your approximation for stresses?</font>
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<font face="arial,helvetica,sans-serif" size="2">2) When you talk about one penalty coefficient, do you mean one lagrange multipliers? </font>
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<font face="arial,helvetica,sans-serif" size="2">3) Are you using Hellinger-Reissner variational principle? I don't see why you also need to use lagrange multipliers to enforce Neumman boundary conditions. Can you explain more on that. I just try to follow the u-sigma formulation (or Hellinger-Reissner variational principle) in Zienkiewicz's book and I don't think that you need to do that for Neumann BCs. </font>
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<font size="2">Alejandro.</font>
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</ul>Tue, 17 Mar 2009 06:18:10 +0000Alejandro Ortiz-Bernardincomment 10131 at http://imechanica.orgPenalty coefficient & max-ent in meshless with mixed formulation
http://imechanica.org/comment/10121#comment-10121
<a id="comment-10121"></a>
<p><em>In reply to <a href="http://imechanica.org/comment/10085#comment-10085">Re: Constrained Moving Least Squares (CMLS) Method</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p class="MsoNormal">
<font face="Calibri" size="3">Dear Alejandro</font>
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<font face="Calibri" size="3">Thank you very much for creating the comment to my notes on the imposing displacement BC's in meshfree method. </font>
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<font face="Calibri" size="3">My M.Sc thesis is mixed formulation and adaptive refinement in the new meshless method named Discrete Least Square Meshless (DLSM) that introduced by my thesis advisor Prof. M.H. Afshar. In this approach we use MLS approximation to construct shape functions. In mixed formulation displacements and stresses obtained simultaneously, therefore for imposing Dirichlet and Newmann BC’s only one penalty coefficient is used. Now my questions are,</font>
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<font face="Calibri" size="3">1) <span> </span>How determine amount of penalty coefficient?</font>
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<font face="Calibri" size="3">2) Can I use maximum-entropy approximation instead of MLS for imposing BC’s without penalty coefficient in mixed formulation for any meshfree method?</font>
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<font face="Calibri" size="3">Thank you</font>
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<font face="Calibri" size="3">J. Amani</font>
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</ul>Mon, 16 Mar 2009 07:15:36 +0000Jafarcomment 10121 at http://imechanica.orgRe: Constrained Moving Least Squares (CMLS) Method
http://imechanica.org/comment/10085#comment-10085
<a id="comment-10085"></a>
<p><em>In reply to <a href="http://imechanica.org/node/4977">Constrained Moving Least Squares (CMLS) Method</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
<font size="2">Hello Jafar,</font>
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<font size="2">I don't know details on the method you ask but if it is similar to MLS and you only need linear approximation I would suggest that you search for Maximum-Entropy approach. It would be more efficient from the computational point of view and you can apply essential boundary conditions exactly as in FEM. You can get some information in these posts:</font>
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<a href="http://www.imechanica.org/node/1215"><font size="2">http://www.imechanica.org/node/1215</font></a>
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<a href="http://www.imechanica.org/node/402"><font size="2">http://www.imechanica.org/node/402</font></a>
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<font size="2"><a href="http://www.imechanica.org/node/402">http://www.imechanica.org/node/3424</a></font>
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<font size="2"><a href="http://www.imechanica.org/node/608">http://www.imechanica.org/node/608 </a></font>
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<font size="2">Best,</font>
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<font size="2">Alejandro A. Ortiz</font>
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</ul>Fri, 13 Mar 2009 05:39:52 +0000Alejandro Ortiz-Bernardincomment 10085 at http://imechanica.org