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Re: [Libmesh-users] Integration points through the thickness for shell elements
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From: Yuxiang W. <yw...@vi...> - 2018-11-18 07:28:32
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Hi all, To follow up my previous email, I think I can simplify my question as below: It seems to me that if we have more than one integration point across the shell thickness, the entire integration would need to be done by manually constructing the Jacobian matrix in 3D to get the JxW values, without using the 2D JxW entry in the current QUADSHELL4 element. This is because this Jacobian in 2D does not take into account the nodal thicknesses and directors, and there are no simple formula to use this information to reconstruct the 3D JxW value. I tried to put these details in the attached PDF in the previous email. If you wouldn't mind, could you please share with me your thoughts on whether my statement above is true? Or that I am missing something? Again, I really appreciate your time to help me on this issue. Best, Shawn -- Yuxiang "Shawn" Wang, PhD yw...@vi... +1 (434) 284-0836 On Fri, Nov 16, 2018, 22:56 Yuxiang Wang <yw...@vi... wrote: > ( I am not sure whether the equations will display fine, so I also > attached a PDF version of the text.) > > > > Hi all, > > > > Thank you all so much for the timely & quality responses! I really > appreciate it. And my apologies for the delayed response. > > > > John – sorry that I did not explain myself clearly. I was using the > notation where the mapping usually is from a geometry with thickness from > physical domain to the reference domain , although shape functions and > not a function of the third reference variable . As a result, I will > still be integrating over all three axis. > > > > Ben & David - thank you so much again for the guidance. That said, I > couldn’t figure out how to do the integration by adding a loop outside the > existing quadrature point loop while utilizing existing JxW values (or more > specifically, the determinant part). I only know how to do that manually > from scratch (i.e., calculate the Jacobian matrix, and manually calculate > its determinant, times the quadrature weight). Would you kindly share > whether you know a paper or textbook talking about this? > > > > My confusion, listed in detail, is below: > > For convenience I am using the notation in the Bathe textbook (which is > consistent with his MITC4 original paper), where he used to denote the > shape functions of the node (sometimes also referred to as or in other > references), and to denote the coordinates in the reference domain > (sometimes also referred to as in other references). > > Intuitively, I think I can do this: > > Where is the full 3D Jacobian determinant and the is the surface > Jacobian determinant that libmesh currently provides for QUADSHELL4, and is > the nodal thicknesses of node . However, I cannot mathematically prove > this equation. > > > > For a general shell element, the coordinates of a point is given by > > The variable denotes spatial dimensions of and denotes the local node > id. Then, is the i-th component of the unit nodal director at node , and is > the nodal thicknesses of node in the direction. Note that the shape > functions only and is not a function of . > > Then, in general, the Jacobian matrix is , > > And we can note the three columns as and (the curvilinear basis for the > convected coordinate system). Also, . > > In the mean time, the Jacobian for the 2D element is > > And if we denote the two columns as then if you don’t mind, I can write > as . > > > > I went back to my intuition equation, the left hand side would be > > And the right hand side would be > > I couldn’t figure how whether in general . Intuitively, it seems to me > that for variable thickness shells where , this equation does not hold. > > > > Thank you for your patience in reading through, and please let me know if > my write-up is not clear enough. Any thoughts would be appreciated! > > > > Best, > > Shawn > > > > On Wed, Nov 14, 2018 at 8:52 AM David Knezevic <dav...@ak...> > wrote: > >> On Wed, Nov 14, 2018 at 11:45 AM Benjamin W. Spencer via Libmesh-users < >> lib...@li...> wrote: >> >>> It's pretty standard for shell elements to have multiple integration >>> points through the thickness at every in-plane integration point. >>> Integrating the response through the thickness allows you to represent the >>> variation in the nonlinear constitutive response of the material through >>> the cross-section, and come up with resultant quantities at the locations >>> of the in-plane integration points, which are then integrated using >>> standard procedures. >>> >>> I haven't really gotten too far into this yet, but I don't think >>> accommodating those extra integration points would involve changing how the >>> integration rules or data structures would work in libMesh. I think you >>> would just evaluate vectors of properties at the standard integration >>> points, with each entry in the vector representing a different point >>> through the thickness. We are just getting started on the path of >>> developing shell elements in MOOSE, so our group will be looking into how >>> to handle this. >>> >> >> Yes, I agree with this description. We do this for modeling composites >> for example, since they have different properties in each layer of the >> composite which you can model via quadrature through the thickness. >> >> However, I think the libMesh example that is being discussed here is just >> meant to be as simple as possible and hence it doesn't do this. Also, I >> believe you can use analytical formulas for the integration through the >> thickness in the case that the material is uniform, so I guess that is what >> is done in the example. >> >> Best, >> David >> >> >> >>> On 11/14/18, 8:19 AM, "John Peterson" <jwp...@gm...> wrote: >>> >>> On Tue, Nov 13, 2018 at 11:22 PM Yuxiang Wang <yw...@vi...> >>> wrote: >>> >>> > Dear all, >>> > >>> > As one usually reads from literature (or commercial software >>> > documentation), usually, a shell element would need >= 2 Gaussian >>> > quadrature points through the thickness to capture its bending >>> behavior. >>> > For example, in the LS-DYNA documentation >>> > < >>> https://urldefense.proofpoint.com/v2/url?u=https-3A__www.dynasupport.com_tutorial_ls-2Ddyna-2Dusers-2Dguide_elements&d=DwICAg&c=54IZrppPQZKX9mLzcGdPfFD1hxrcB__aEkJFOKJFd00&r=hn5akMybrkn-1oiQB8nm_y7trT_BOQm9jBgbzQWwxXA&m=d16wjsgwuY6Xejdr47KKgE8srFi-kHjT92yv6KeNbt0&s=XxuqMBzy7V7pzqu5KNGEyqWuMwh-JQEAJerwVZsdQFU&e=> >>> or >>> > mentioned in this paper >>> > < >>> > >>> https://urldefense.proofpoint.com/v2/url?u=http-3A__web.mit.edu_kjb_www_Principal-5FPublications_Performance-5Fof-5Fthe-5FMITC3-2B-5Fand-5FMITC4-2B-5Fshell-5Felements-5Fin-5Fwidely-5Fused-5Fbenchmark-5Fproblems.pdf&d=DwICAg&c=54IZrppPQZKX9mLzcGdPfFD1hxrcB__aEkJFOKJFd00&r=hn5akMybrkn-1oiQB8nm_y7trT_BOQm9jBgbzQWwxXA&m=d16wjsgwuY6Xejdr47KKgE8srFi-kHjT92yv6KeNbt0&s=tSn1-9z_P8T0uME2jwoYDUO7AQEElPc8f3IjxytF-EA&e= >>> > > >>> > . >>> > >>> >>> I guess I'm confused about what you mean by "thickness". Our SHELL >>> elements are logically two-dimensional (have zero thickness) so IMO >>> it >>> doesn't make sense ask about integration in the transverse >>> direction... >>> >>> -- >>> John >>> >>> _______________________________________________ >>> Libmesh-users mailing list >>> Lib...@li... >>> >>> https://urldefense.proofpoint.com/v2/url?u=https-3A__lists.sourceforge.net_lists_listinfo_libmesh-2Dusers&d=DwICAg&c=54IZrppPQZKX9mLzcGdPfFD1hxrcB__aEkJFOKJFd00&r=hn5akMybrkn-1oiQB8nm_y7trT_BOQm9jBgbzQWwxXA&m=d16wjsgwuY6Xejdr47KKgE8srFi-kHjT92yv6KeNbt0&s=CrRFwn9nab2x8xz4ZBh1T-gH5-hRr1_hZ-mFsYImu-E&e= >>> >>> >>> >>> _______________________________________________ >>> Libmesh-users mailing list >>> Lib...@li... >>> https://lists.sourceforge.net/lists/listinfo/libmesh-users >>> >> > > -- > Yuxiang "Shawn" Wang, PhD > yw...@vi... > +1 (434) 284-0836 > |
