Discussion Forum

Posted: 1 decade ago 2013年4月8日 GMT-4 08:43

Hi the easiest is to add your "volume" as a domain, but exclude it from the SOLID, then add ALE as first physics node, apply it to the domain (free displacement), drive the interface boundary by displacements u,v,w from solid and calculate. then Results Derived values integrate operand 1 over this pure math ALE volume. Another way, not so clean, add a very "soft" material (E=1[Pa], nu =0, rho = 1 for the volume you want to integrate, calculate everything under Solid, then integrate over the deformed volume (spatial frame) for the "soft material" -- Good luck Ivar

Posted: 1 decade ago 2013年4月8日 GMT-4 17:47

Hi, In most cases you can perform a volume integral without actually meshing the volume. The trick is to use the divergence theorem to transform the volume integral to a surface integral. [math]\int_V{1} dV = \int_A{x n_x} dA [/math] For an example, look at the model hyperelastic_seal. Note that the integral must be taken on the spatial frame, and with current x and deformed normal. Regards, Henrik

Posted: 1 decade ago 2013年4月9日 GMT-4 06:44

Hi Henrik Thankls for the tip, indeed it avoids meshing everything ... -- Good luck Ivar

Posted: 1 decade ago 2013年4月11日 GMT-4 16:31

Hi Henrik, I have though and tested a little your example, I find you are too restrictive, that works OK for "simple cube examples" but can easily fail for more complex models, and you should comment on with or without geometrical non-linearities, as then the reference frames can easily play us some games ;) My suggestion is to use the full 3D (or 2D respectively) representation of the divergence theorem and state that [math] \iiint_V \bold{\nabla\cdot(x,y,z)} \:dV = 3 \:Vol = \int\!\!\!\!\!\!\bigcirc\!\!\!\!\!\!\int_A \bold{(x,y,z)\cdot n} \:dA = \int\!\!\!\!\!\!\bigcirc\!\!\!\!\!\!\int_A (x \; n_x + y\; n_y + z \; n_z ) \:dA [/math] respectively [math] \iint_A \bold{\nabla\cdot(x,y)}\:dA = 2\:Area = \oint_l \bold{(x,y)\cdot n}\:ds = \oint_l (x \; n_x + y\; n_y)\:ds [/math] This is OK for closed surfaces, it becomes trickier for just an open part, as one cannot make the full closed boundary integral, then one should consider, the changed volume via the (u,v,w) vector (assuming the non represented part of the closed integration is fixed in space) [math] \Delta Vol = \iint_A (u\;n_x+v\;n_y+w\;n_z)\:dA [/math] or simply in 2D (and add some 2*pi*r and change names for 2D-axi) [math] \Delta Area = \int_l (u\;n_x+v\;n_y)\:ds [/math] Then comes the story of the frames and when these are dissociating (x,y,z) and (X,Y,Z) (spatial to material) for geometrical non-linearities, not to forget sign convention for normal directions. In traditional linear mode, the integration of (x*nx ...) will not give the changing volume, I notice one can only get it with the complement of the closed volume plus the integration of the (u*nx+...) With non linearities the integration of (x*nx ...) is rather correct in closed integration, partly in open integration all depends on shapes, and one can easily get double effects if one mix up the frames for integration, but the method of the complement of the volume with (u*nx ...) is rather wrong because often your get easily side effects as in the attached examples. Finally, to come back to your model library example, it works nicely for compressible fluid enclosed in a full closed volume, but I have been fighting with a similar example where I have water (incompressible) fluid in the closed volume and not "air", there this approach gives me only a stiff solution and the solver fails, so far. Update: with some tweaking I managed to get the library model to work with the bulk modulus of water and a gamma of 1 instead of a pressure of 1[atm] with a gamma of 1.4. (and for some reason, the last months the COSMOL Forum used from an Opera Browser does not allow to upload files other than jpeg ? have to fight with my IE...) -- Good luck Ivar

Attachments:

• MeshInt_5.mph
• MeshInt_4.mph

Posted: 1 decade ago 2013年4月12日 GMT-4 08:59

Interesting discussion. Ivar I don’t think the equation you provided for change in volume/area in terms of displacements is correct. If you set x=X+u in the original equation and expand you will see that it is only valid when nx does not change. Therefore it is only valid for infinitesimal displacements. Nagi Elabbasi Veryst Engineering

Posted: 1 decade ago 2013年4月12日 GMT-4 10:14

Hi Nagi Exactly that is also what I see in my attached models but it's correct, and the only way, with geometrical non-linearities turned off ! Probably I should then rather write u*nX+v*nY... for that case. The difference is, as you pointed out there, when you turn on the the geometrical non linearities that my attempt with the equation with u*nx ... ARE INDEED WRONG. From the moment the following equality is NOT TRUE the two values diverges. But for simple "cubic" cases, this equality can often be true, so I did first conclude it was OK on a special case (hence newer trust a simple cubic model in Cartesian coordinates ;): [math] \int\!\!\!\!\!\!\bigcirc\!\!\!\!\!\!\int_A (X\;n_X+Y\;n_Y+Z\;n_Z)\:dA = \int\!\!\!\!\!\!\bigcirc\!\!\!\!\!\!\int_A (x\;n_X+y\;n_Y+z\;n_Z) \:dA [/math] (generally wrong in spatial frame) All this for me is to try to find an easy way to estimate the volume change of a closed volume (i.e. a volume under a membrane) from the deformations of the membrane, without modelling the full closed volume. It's typically for simple modelling of a closed fluid (incompressible liquid) that interacts with a structure, without going through CFD. In fact after some tweaking I managed to get the model library compressing seal model to converge, by replacing the "air" inside the seal by an equivalence water volume that remained constant. And all this ultimately because I'm looking for a simpler way to model the bio-mechanics of an eye ;) -- Good luck Ivar

Posted: 1 decade ago 2013年4月12日 GMT-4 10:54

Hi Ivar, I will probably use this quote from you in the future “never trust a simple cubic model in Cartesian coordinates” :) I’m involved in eye modeling too, and it’s a challenge to develop accurate relatively simple models that get the physics right. Nagi Elabbasi Veryst Engineering

Posted: 1 decade ago 2013年4月14日 GMT-4 11:49

Hi, I have some comments: 1. If you write the expression to integrate as (X+u)*nx and put the integration on spatial frame (the default), it will work with both with geometric nonlinearity on and off. In the case of geometric nonlinearity this is correct since x=X+u, and nx is the deformed normal. In the geometrically linear case, then the displacements are assumed to be infinitesimal. So it should not matter whether the integration is on material or spatial frame (they coincide). Also, in the first approximation, there is no difference between the undeformed normal nX and the deformed normal nx. 2. Whether to use x*nx, (x*nx+y*ny)/2 or any similar expression is not really important. But in some cases it is possible to save computational work by doing a smart choice. If you e.g. have a symmetry plane in the x-direction, then ny=nz =0 there. So by using y*ny or z*nz you could exclude the symmetry plane from the integral. 3. If you are not only using the integral for evaluating the volume, but actually feed the data back into some equation, then it will (at least in 3D) often be necessary to use the nojac() operator. The stiffness matrix will connect all the boundary degrees of freedom, and the chance of running out of memory is large. Using the nojac() operator will slow down the convergence rate since the stiffness matrix is no longer optimal. Regards, Henrik

Posted: 1 decade ago 2013年4月15日 GMT-4 00:57

Hi Henrik Thanks for the precisions, and I agree your point 2+3 are important for solving time / convergence with the equations, but one must choose the correct direction, as my first example was constant in "x" and so I did not get any volume change, therefore my generalisation for x,y,z. But from the test I did further and specially for general closed volumes there is hardly any difference for x,y,z taken separately. So its better to know and state clearly the hypothesis, then to apply the best choice -- Good luck Ivar