iMechanica - Comments for "Journal Club Theme of September 2014: Numerical modeling of thermo-hydro-mechanical coupling processes in porous media"
https://imechanica.org/node/17096
Comments for "Journal Club Theme of September 2014: Numerical modeling of thermo-hydro-mechanical coupling processes in porous media"eninjective
https://imechanica.org/comment/26563#comment-26563
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<p><em>In reply to <a href="https://imechanica.org/comment/26398#comment-26398">Thanks for your explanation!</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>Hi Jinxiong, </p>
<p>I want to add one more point. While inf-sup conditon implies that the matrix B^{T} is injective, injective of B^{T} does not imply that inf-condition holds. In fact, when we consider stability of a given numerical method, we need to consider a sequence of matrices A and B corresponding to a sequence of meshs with mesh size approaching zero. The inf-sup condition is a necessary (but not sufficient) condition that requires the exisits of a positive constant beta<strong> that is independent of meshsize h, </strong>such that </p>
<p>x^{T} B^{T} y \geq beta ||x|| ||y||</p>
<p>This is a more strict requirement than the injectivity of B^{T} </p>
<p>Thanks,</p>
<p>WaiChing</p>
<p> </p>
<p> </p>
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</ul>Tue, 11 Nov 2014 04:19:39 +0000WaiChing Suncomment 26563 at https://imechanica.orgLooking forward to read it.
https://imechanica.org/comment/26403#comment-26403
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<p><em>In reply to <a href="https://imechanica.org/comment/26401#comment-26401">A future review article in Advances in Applied Mechanics</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>Hi Stéphane,</span></p>
<p><span>Thanks for pointing out the article to me. I would love to take a look when the book is available. </span></p>
<p><span>BTW, I wonder if you have encounter similar inf-sup condition issue in isogeometric models especiually when fomrulating something that leads to a block system (e.g. incompressible elasticity with displacement and hydroastatic pressure as nodal solution)? </span><span>Of course the basis functions are different between the conventional FEM and isogeometric Galerkin. A</span><span>ny important literature we should be aware of?</span></p>
<p><span>Best Regards,</span></p>
<p>WaiChing</p>
<p><span> </span></p>
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</ul>Thu, 18 Sep 2014 14:22:28 +0000WaiChing Suncomment 26403 at https://imechanica.orgA future review article in Advances in Applied Mechanics
https://imechanica.org/comment/26401#comment-26401
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<p><em>In reply to <a href="https://imechanica.org/node/17096">Journal Club Theme of September 2014: Numerical modeling of thermo-hydro-mechanical coupling processes in porous media</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>Dear WaiChing, Thank you for this very interesting theme. I wanted to let you know that a paper reviewing recent advances in THM fracture will be appearing soon in Advances in Applied Mechanics. <a href="http://www.elsevier.com/books/book-series/advances-in-applied-mechanics">http://www.elsevier.com/books/book-series/advances-in-applied-mechanics</a> Best regards,</p>
<p>Stéphane <a href="http://wwwen.uni.lu/recherche/fstc/research_unit_in_engineering_science_rues/members/stephane_bordas">http://wwwen.uni.lu/recherche/fstc/research_unit_in_engineering_science_...</a></p>
<p><a href="http://handle.net/handle/10993/11024">http://handle.net/handle/10993/11024</a></p>
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</ul>Thu, 18 Sep 2014 08:12:29 +0000Stephane Bordascomment 26401 at https://imechanica.orgThanks for your explanation!
https://imechanica.org/comment/26398#comment-26398
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<p><em>In reply to <a href="https://imechanica.org/comment/26376#comment-26376">swelling gels and inf-sup condition</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>Dear WaiChing, Thanks for your detailed explanation! I learn a lot through reading your post. Jinxiong</p>
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</ul>Wed, 17 Sep 2014 06:24:57 +0000Jinxiong Zhoucomment 26398 at https://imechanica.orginf-sup condition
https://imechanica.org/comment/26386#comment-26386
<a id="comment-26386"></a>
<p><em>In reply to <a href="https://imechanica.org/node/17096">Journal Club Theme of September 2014: Numerical modeling of thermo-hydro-mechanical coupling processes in porous media</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>Another new application for inf-sup condition is for concurrently coupling models across different scales. In particular, Bauman et al 2008 has conducted a detailed analysis on the inf-sup condition of Arlequin method that couples particle and continuum models in 1D. </p>
<p><span>Bauman, P. T., Dhia, H. B., Elkhodja, N., Oden, J. T., & Prudhomme, S. (2008). On the application of the Arlequin method to the coupling of particle and continuum models. </span><em>Computational mechanics</em><span>, </span><em>42</em><span>(4), 511-530.</span></p>
<p>In addition to the mathematical analysis, the inf-sup condition can also be checked numerically via the inf-sup test. The inf-sup test is just a generalized eigenvalue problem corresponding to the inf-sup condition. I have done some work with Dr. Alejandro Mota from Sandia in which we construct an inf-sup test for a multiscale model that couples local and non-local constitutive responses. </p>
<p><span>Sun, W., & Mota, A. (2014). A multiscale overlapped coupling formulation for large-deformation strain localization. </span><em>Computational Mechanics</em><span>,</span><em>54</em><span>(3), 891-891.</span></p>
<p>It is interesting to note that both multiphysics AND multiscale problems might lead to block system that requires the finite dimensional spaces used to interpolate different variables being compatible and thus the inf-sup condition. </p>
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</ul>Mon, 15 Sep 2014 04:47:57 +0000WaiChing Suncomment 26386 at https://imechanica.orgswelling gels and inf-sup condition
https://imechanica.org/comment/26376#comment-26376
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<p><em>In reply to <a href="https://imechanica.org/comment/26369#comment-26369">inf-sub condition?</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>Hi Jinxing, </p>
<p>Yes, I do think that a lots of the numerical schemes developed for THM problem are appliable to other multiphysics problems. Of course, the subtle difference for various problems are important. In fact if we look at the drained and undrained split slovers by Armero, it really resembles the adiabatic and isothermal solvers developed by Simo and Miehe in the 90s. </p>
<p>Now regarding the inf-sup condition (often referred as Ladyzhenskaya-Babuska-Brezzi or LBB condition), it is basically a necessary condition for spatial stability. Let me try to explain it in the most simplistic way I could. Note that there are many papers discussing this topic --- I do not think I am the most qualified one to answer this question and hence I would appreciate it if any audience/reader can point out any error or mistake in my explanation. </p>
<p>First, consider a mutliphysics governing equation that leads to a matrix form that reads (sorry I dont know how to write the xml):</p>
<p>[A B^T ; B C] [u; p] = [f g]</p>
<p>For poromechanics probelm, the most difficult part is when the pore fluid diffusion is negligible (i.e. near the undrained limit), in which case, we have (assueming incompressible constituents),</p>
<p>[A B^{T; B O] [u; p] = [f 0]</p>
<p>In this case, the material is undrained and the pore-fluid constituent will provide an incompressible constraint to the solid skeleton. One thing we need to pay attention is that B matrix that controls the two-way coupling is not a square matrix but a rectangular one. Now, assume that A is positive definite. We can use the Schur complement B A^{-1} B^T to solve for p, i.e. ,</p>
<p>B A ^-1 B^T p = B A^-1 f</p>
<p>However, if the null space of (B^T) is spanned by any <em><strong>nontrivial</strong> </em>basis, then there may exist a column vector with non-zero components (let's called it lambda) that satisfies A^-1 B^T lambda = 0. That may cause spatial stability issue because a spatial oscillatiing (checkerboard pattern) pore pressure field may occur (due to the existence of such a column vector), but we have no "control" over it (i.e. u = A^-1 B^T lambda is zero but lambda is not). In other words, the entire block system becomes ill-posed. </p>
<p>It is proved that such oscillation occurs when the pore pressure and displacement are discretized by the same set of basis function for the saturated poromechanics problem or other mathematically similar systems (Darcy's flow with V-p pair, incompressible elasticity with u-p pair, Stokes' problem with v-p pair...etc). There are multiple ways to avoid that. For instance, one can modifies the B matrix or add something in the C matrix to elimate the osillation mode. </p>
<p>Below is an example in which spatial oscillation occurs near the undrained limit in a poro-elastic beam (RIGHT), and the stable solution obtained via a project based stabilized finite element model (LEFT). I have included more examples in my own papers [62,63] and there are many more in the literature. </p>
<p><img src="http://poromechanics.weebly.com/uploads/2/2/9/7/22975762/1410459319.png" alt="" width="637" height="252" /></p>
<p>I must confess that my explanation is quite shallow. Below is a sample of work concerning the inf-sup condition:</p>
<p><span>Bathe, Klaus-Jürgen. "The inf–sup condition and its evaluation for mixed finite element methods." </span><em>Computers & structures</em><span> 79.2 (2001): 243-252.</span></p>
<p><span>Dohrmann, Clark R., and Pavel B. Bochev. "A stabilized finite element method for the Stokes problem based on polynomial pressure projections." </span><em>International Journal for Numerical Methods in Fluids</em><span> 46.2 (2004): 183-201.</span></p>
<p><span>Fortin, Michel, and F. Brezzi. </span><em>Mixed and hybrid finite element methods</em><span>. New York: Springer-Verlag, 1991.</span> </p>
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</ul>Thu, 11 Sep 2014 18:43:53 +0000WaiChing Suncomment 26376 at https://imechanica.orginf-sub condition?
https://imechanica.org/comment/26369#comment-26369
<a id="comment-26369"></a>
<p><em>In reply to <a href="https://imechanica.org/node/17096">Journal Club Theme of September 2014: Numerical modeling of thermo-hydro-mechanical coupling processes in porous media</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>Dear WaiChing, Thank you very much for posting this very interesting topic. A related topic is the modeling of swelling of polymer gels, where the theory of poroelasticity is also used. For gels, we have, typically, two sets of indepedent variables at each node, the chemical potential of fluid and the displacement. I think most of the scheme and procedure for THM is also applicable for modeling gels, albeit some modification is needed. I would like to have your comment on this point.</p>
<p>BTW, can you explain in basic language what is inf-sub condition? What's its effect on numerical stability?</p>
<p>Jinxiong</p>
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</ul>Thu, 11 Sep 2014 00:51:42 +0000Jinxiong Zhoucomment 26369 at https://imechanica.orgCoupling finite volume fluid solver with FEM solid solver
https://imechanica.org/comment/26360#comment-26360
<a id="comment-26360"></a>
<p><em>In reply to <a href="https://imechanica.org/node/17096">Journal Club Theme of September 2014: Numerical modeling of thermo-hydro-mechanical coupling processes in porous media</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>I also want to add the FEM-FVM coupling model derived by Pr. Prevost's research group at Prinecton and some of its application on CO2 storage problem. This type of multi-model coupling approach is useful for the following reason --- many commerical or research reservoir simulators are written via a finite volume approach, while solid solvers are often written in finite element. While one can always "reinvest the wheel" to make a pure FVM or FEM multiphysics solver, it might be easier (and cheaper) to establish a FVM-FEM coupling if the uncoupled solid and fluid solvers are available to the modelers. </p>
<p><span>“Two-way coupling in reservoir–geomechanical models: vertex-centered Galerkin geomechanical model cell-centered and vertex-centered finite volume reservoir models”. J.H. Prévost. </span><em>International Journal for Numerical Methods in Engineering</em><span>. 2014. DOI: </span><a class="liexternal" href="http://dx.doi.org/10.1002/nme.4657" target="_blank">10.1002/nme.4657</a><span>.</span></p>
<p>The second paper is on studying the thermal stresses of the caprock. Notice that when CO2 is injected underground, it is often injected in its super-critical form, i.e., a combination of pressure and temperture that keeps the CO2 in s dysyr midway between a gas and a liquid. Of course, the CO2 will not have the same temperture of the acquifer or the cap rock and that temperature difference and the bulid up of pressure plume due to the injection may be sufficient enough to affect the integrity of the cap rock.</p>
<p>One thing I notice in this paper is that there is a structural heating term in the balance of energy from both the solid and fluid constituent, which is neglected by numerous previous work. This of course will make the coupling easier to model as it reduces the THM from a three-way coupling problem to a two-way coupling one (i.e. one can solve for the temperature first and then update the displacement and pore pressure without losing accuracy). I also wonder what is the advantage of putting the degree of saturation as the nodal solution? It seems like it may make the THM problem more complicated especially considering the fact that here may be a "degree of saturation jump" between layrs that have profoundly different wettiability. </p>
<p><span>“Effect of thermal stresses on caprock integrity during CO2 storage”. G.Y. Gor, T.R. Elliot, J.H. Prevost.</span><em> International Journal of Greenhouse Gas Control</em><span>. 12. 2013. pp. 300-309. DOI:</span><a class="liexternal" href="http://dx.doi.org/10.1016/j.ijggc.2012.11.020" target="_blank">10.1016/j.ijggc.2012.11.020</a><span>.</span></p>
<p> </p>
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</ul>Mon, 08 Sep 2014 01:43:04 +0000WaiChing Suncomment 26360 at https://imechanica.orgIterative scheme
https://imechanica.org/comment/26359#comment-26359
<a id="comment-26359"></a>
<p><em>In reply to <a href="https://imechanica.org/node/17096">Journal Club Theme of September 2014: Numerical modeling of thermo-hydro-mechanical coupling processes in porous media</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>Hi Steve,</p>
<p>Your discussion is interesting. I have included a recent paper that I have found through a web search:</p>
<p>Y. Xie+, G. Wang*. “A Stabilized Iterative Scheme for Coupled Hydro-mechanical Systems Using Reproducing Kernel Particle Method”, International Journal for Numerical Methods in Engineering, Vol. 99 (11), 819-843. DOI: 10.1002/nme.4704.</p>
<p>A link to the paper can be found here: <a href="http://ihome.ust.hk/~gwang/Publications/XieWangIJNME2014.pdf">http://ihome.ust.hk/~gwang/Publications/XieWangIJNME2014.pdf</a></p>
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</ul>Mon, 08 Sep 2014 00:42:41 +0000kwlimcomment 26359 at https://imechanica.org