A variational approach to fluid-structure interaction
Benešová, Barbora
Kampschulte, Malte
Schwarzacher, Sebastian
In this talk we consider the interaction of a Stokes/Navier-Stokes flow with a viscoelastic body. The elastic body is allowed to undergo large deformations (but no self-collisions). In order to handle this situation correctly, we devise a variational approximation scheme in the spirit of DeGiorgi to the combined problem. Moreover, by using a two-scale scheme, we also extend this approach to the hyperbolic regime including inertia of the solid body. These variational approaches allow us to prove proper energetic estimates while also controling the geometric restictions posed on the solid body and, eventually, to establish existence of weak solutions.
 I want to introduce a variation in approach to fluid structure interaction. And this is work which was done in collaboration with Malte Kampschute and Sebastian Schwarzacher, who are also both from Prague. So let me just start what I want to talk about. So what I want to talk about is fluid structure interaction. So I mean, the world almost tells it everything. So fluid structure interaction happens always when we have two materials. One is kind of a fluid material and one is a solid material. And they both move and have mutual movement somehow. They influence each other on the movement. So for example, I took this example here from some Journal of Computational Physics. But of course, there are many examples and you can think of like a fluid which is flowing next to an elastic membrane. So this thing here is kind of an elastic membrane. And as the fluid flows, of course, it exerts forces onto this elastic membrane and it starts to move. So in this example, which I just took from this Journal of Computational Physics, it goes up and down and up and down. So this elastic membrane moves quite a lot. So what you can see already that there are many examples in which you have an interaction between fluids and solids. And we can also see that the solid is really moving and deforming a lot. So what you can see is that the deformation of the solid can be large. And it shouldn't be enough to say, OK, we think only of some small or rigid bodies because really here you see an example where the deformation can be quite large. OK, so that's what we want to consider. So maybe that's the main thing of what will be the talk. We will have a fluid structure interaction. We will have some fluid and we have a solid which can really have large deformations. So let me just go to the model and let me just tell you what the key ingredients of this will be, which we have. So we will, of course, fix everything to some fixed container. So you can think of a fixed container like this. So think here, this is like what happens at time zero or what I think of as a reference configuration. So this gray part here is my solid part, I think. And the solid part is best described by its deformation, which I hear the note by Ita. Because of course, we are mostly, for the solid, we are mostly interested where it is. I mean, it's not so important how fast it moves. It's really important where it is. So therefore, we use this deformation here. For the fluid, on the other hand, the main variable which describes what it does is not really where it is because that's not so important for the fluid as it flows all the time. But the main variable here is the velocity. So that's also clear. So what we have is a coupling between Lagrangian and Eulerian coordinates because of course our solid is really described by this deformation, which you can see here. It tells you where every point goes to and the solid by its velocity. So once I have these key ingredients, which describe my materials, I can start up, set up my modeling and for my modeling, I will just take the basic physical laws and those should be satisfied. So the basic physical laws are the balance of masses and momentum. So we have the balance of mass and momentum for the fluid. So that's of course, I mean, I could take a lot of possibilities for the balance of mass and momentum, but I don't think about it a lot. And I just take Navier-Stokes equation, which is maybe the easiest model I can think of for a fluid and I take just the Newtonian fluid. So I have here the inertial terms. I have here the diffusion, which I will call dissipation. I have some pressure and I have some forces and the divergence is zero because everything is of course incompressible. And for the solid, I have the balance of course also of mass, but this is not so important. And I also have the balance of momentum, which here reads, okay, I have the inertial term. I have some divergence of some pressure of some stress tensor sigma, which I will specify in a minute, what is this stress tensor sigma. So this is going to be like a modeling issue and they should be equal to the forces. So it's also Newton's second law. It's just the balance of momentum, nothing special here. And of course, over the mutual boundary, which you hear, this is the mutual boundary between the fluid and the solid, there should be some transition conditions. These transition conditions tell you really how the influence between the two materials goes on. And in our case, we take maybe the most simple transition conditions we can think of. We take the continuity of the velocity over this common boundary, which somehow in the microscopical setting means that all the atoms move as the same fast. And also, we take a continuity of traction, so it means the force over this boundary is continuous. So this here, the first here is really the continuity of velocity, which is clear, but it's written down in this reference configuration because only there we can talk about the solid velocity. And the second here is, you already see, this is the stress tensor applied to the normal, so this is the forces. And here you just have the stress tensor stemming from the Navier-Stokes equation, revered and again to the reference configuration. So this is the boundary terms. And now, okay, so you could say this is a complete model, but it's not because what I'm missing here is really the stress tensor for the solid. So the stress tensor for the solid needs to encode some modeling, and we will put a little bit more complexity here into the solid or allow the stress tensor to be a little bit more complex, particularly because I said that we want to have a look at these large deformations. It's okay. We want to have large deformations. So if you want to do that, then of course, the formation of the solid somehow specifies the boundary of the main or the domain of definition for the fluid. So of course, the deformation of the solid cannot be arbitrary. In fact, in order to set everything up and be in the correct setting, the deformation needs to be homomorphism at least. So we need to choose our stress tensor rightly. In fact, from the point of view of modeling solid materials, this is not like something additional we ask for because when modeling solid materials, asking that formations to be homomorphisms is something really standard or one should do this all the time. So we will of course not consider all the materials. So we will consider only such materials for which the stress tensor is given by two potentials. So one of them is like an energy potential and one of them is like a dissipation potential. So there are really many materials which fall into this class. And I guess similar concept is known in fluid mechanics also, but in solid mechanics, this is most often called the concept of generalized standard materials going back here to half an a million. And this is a concept which gives you like really a wide class of materials which you can think of. And for the talk here, I will further just think of a prototypical example for this energy and dissipations which you want to consider. So let me start with the energy here. The prototypical energy I want to consider looks like this complicated expression here. So let me just talk you a little bit through what I want to have. So the first term here is just a norm. So it tells you how far the gradient more or less is from the identity. So it's an elastic term and here's like the tens of elastic constants. So it's actually the standard Sanctum and Kyrchov energy. So it really just tells me that my material actually would like to be relaxed and is really in a stress-free state once there are no forces. Once it's really an identity mapping here, the deformation gradient, so it's really not the form at all. We also consider some two additional terms. So the first additional term I put here is like inverse of the determinant of the gradient of the formation. So the inverse of the Jacobian of the deformation. And this is really a term which is needed and physical because once you start considering large deformations, it never can be that you could compress a finite amount of the material to zero volume with a finite energy. This should never happen. This should always be penalized. So it's clear that any physically reasonable energy should blow up to infinity as the Jacobian of the deformation goes to zero. And here we just do it in this term. And as a last thing, I consider a regularizing term. So I add here the second gradient of the deformation to the energy, which is something which is maybe not standard and which is sometimes called second grade materials. And we do it for mathematical reasons mostly because we need more regularity. Let me just tell you that if we have the second order term, we will know a little bit more about the deformations of finite energy. In particular, we will know that the Jacobian of any such deformation of finite energy will be bounded away from zero. So it will be positive. There will exist some positive epsilon. And we will know that this determinant will be always bigger than some epsilon. This will tell you that this term here, the first term here, will actually not blow up polynomially, but will be bounded, which is of course important. But it will also be important for us from the point of view of many ways we will see some examples in the talk later. Let me just note that this energy overall is non-convex. And it has to be non-convex because we added this Jacobian here. And this is something physical. So if we want to consider large deformations for the solid, we can never take a convex energy and work with a convex energy. Even this tankfrenam-Kirchhoff energy is non-convex, but OK, so this maybe could be convexified. This is maybe not the point. But the point here is that this blow up would never be possible. OK. So this is like the energy I have in mind. And for the dissipation potential, I think of something very simple. I take a quadratic dissipation here. But of course, I have to take it quadratic, not in the deformation gradient, because that would not be physical, because there would not be an independence of the observer. But I have to do it in the Cauchy-Green-Penzo, which I do here. But of course, the downside of it is that the dissipation potential starts to depend on the state. So we will see later why this is maybe something to highlight a little bit. So OK. So this is the modeling I want to consider. So let me just wrap up and have a look at the strong formulation of what I want to do. So I have the following system. I have here my balance of momentum for the solid. So I have here the inertial term. I have here the derivative of the energy, the derivative of the dissipation potential, the forces. I have here the classical Navier-Stokes equation. I have the divergence, which is somehow missing here. The divergence of v equal to 0. I have the transition conditions between the solid and the fluid. And I have further initial and boundary conditions on this omega. OK. So this is the system I want to consider. And what I ultimately want to do, I want to prove existence of weak solutions. At best, we are a suitable time discretization scheme. So why a time discretization scheme? Because, of course, the domain changes all the time because the solid is moving. So time discretization is somehow something which helps because we can fix the domain for a very short time. OK. So this I want to do. But I will just hit OK. But I want, in order to make things simpler, I want to consider a simpler situation. In fact, I want to, for most of the talk, I will, I will, I will, I will, I will, I will consider a situation of slow movement. So what does it mean, slow movement? It means that the movement of the solid and the fluid is so slow that these inertial terms, which are here, are not needed because they are so small, they can be neglected. So this is something which I would call in the, in the, in the physics or in the modeling of solid materials, we call all this quasi-static because it's almost static. The solid is not moving a lot. For the fluid, it's not really static because it's moving, but it's more, more steady flow. So it's like quasi-static or quasi-steady situation. And this is what I want to consider. So these two inertial terms, I drop. And this is the system that we will have a look at most of the talk. OK. So we have here a system which got parabolic, which consists now only of forces, which are here on the right-hand side, the pressure. But OK, so the pressure. And derivatives of energy and dissipation potentials. Exactly. We have here like energy, we have here dissipation potential. And this diffusion here in, in, in the fluid, I can also understand as a derivative of a dissipation potential because of, yeah, it can be understood in the very same way because it plays the same role. OK. So this is the strong system. So let me now just highlight a little bit how the weak formulation of this system looks. So if I want to design a weak formulation for this, of course, I have to take care of these transition conditions. And you would probably agree that the second transition condition here looks the most ugly. So the continuity of tractions. And it is even the most ugly because it features here the stress tensor, which is something which is the least regular thing we know from, from here because, yeah. So it would be best if we somehow could have this condition somehow disappear. And this is what we can do in the weak formulation. In fact, if we take our test functions, so let me just have a guide to your attention at first to the very end of the weak formulation. So if we take our test functions here, here phi is the test function in the solid and psi is the overall test function, which lives on the whole of omega, but in particular is also the test function for the fluid. So if we take them coupled on the solid part in such a way that they depend on the actual solution, then it's kind of a not too hard computation that in case everything is very smooth for such test functions, this equality of tractions comes out automatically. So let me just have a look. So I have, I will call a pair at a weak solution of this parabolic fluid structure interaction problem if it has some regularity, if they still satisfy the first coupling condition. Okay, and if a weak formulation, which actually just consists of transferring here one derivative and that's more or less this and tested by the right test function holds true. So this is the weak formulation I want to consider. And it's really just magical here that this equality of tractions need not to be considered anymore. Okay. So for this weak formulation, I want to prove that from that there exists weak solutions actually according to this week formulation. So this is the theorem. There exists a weak solution, more or less. So let me just tell you how we prove this theorem. So the proof of the theorem is based on some consecutive minimization in the spirit. So we will set up a lot of minimization problems, which are time spent, the time stepping. So sometimes this is called a time incremental minimization, and it's very much used in solid mechanics. So I just send you here and it actually goes back to the spirit of the Georgie or maybe even earlier and is minimizing movements. So let me have a look how this looks like. So I want to set up this minimization schemes, which looks like minimization or minimizing movements. So I fix a time step size here. I start with some initial data. And if I assume that there is some that I have the eta tk given from my previous step, then I construct the next time step as follows. I minimize the energy plus the dissipation needed to go plus the dissipation somehow needed to go from the last step to the step. So I have here, I minimize here the energy. I here take the dissipation potential, but I multiply this by tau. I take here another dissipation potential, which comes from the fluid. And I also add here the forces. So this is what I minimize. And I minimize it subject to eta just in a natural symbol of space, the divergence of v should be equal to zero. And there is a coupling condition on the boundary between which more or less corresponds to this Dirichlet coupling condition, like the equality of velocities we had before. So let me highlight some features of this scheme. So first of all, there is no conditions on the stresses neither here. So we didn't have it in the weak formulation. And we also don't have it in this time stepping scheme, because the equality of stresses comes out automatically from the minimization, because the energy plus dissipation is only minimal if there is really no stress over the boundary and more or less. Let me have a look that the scheme is somehow explicit implicit. So it's implicit here in the energy. This is important because it allows us to control the Jacobian and the geometry of the whole thing. It is implicit here in the v's and it's implicit in the dissipation in the somehow rate variable, which is here the velocity and here also like the discretized velocity. That is explicit in the state variables. So the state variable would be here, the formation and also for the dissipation of the fluid, the state variable would be the domain on which we live. So in fact, during the minimization, all domains are fixed. The domain for the fluid is given somehow from the last step, as well as here the same holds here for the dissipation potential. And this is actually needed in order to get the right Euler Lagrange equation for the solid. This is something which has already been used and for the fluid we needed to modify this accordingly. And also let me say that we have an explicit dependence here in the coupling condition. So when we couple, we say that the velocities should be the same. In order to couple here the velocities, you also have an explicit relation here. And this is important to really get the stress balance now. So once, okay, so now I can, I mean it's standard in the calculus of variations that I can prove existence of minimizers for the scheme. This is really something very, very simple. So once I have existence of minimizers, I can start doing something with them. And the advantage of having a variational scheme is that I can deduce an energy estimate quite easily. If I compare the energy of the minimizer to say the last step and zero, which is something very like most natural thing one would come up with, I already have here an energy imbalance which is the correct one. So I have here that the energy of the current step plus the dissipation, this is all the dissipation here, is bounded by the energy of the last step plus the force times. So in order, so if I now estimate these force terms, which I can do say by, sorry, which I can do by constant qualities here in order to estimate them into this dissipation, I actually get an energy imbalance immediately. So I have here that the energy plus the total integrated over time dissipation is bound by some constant. So this is like an immediate a priori estimate and actually this a priori estimate is more or less everything that is needed to later to go to the to pass to the limit and to the use the existence of weak solutions. Okay, because we know that these energy and dissipations are coercive in the right way. So this is actually already an a priori estimates, which tells you everything. So let me at this point tell you why a variational scheme is so useful. I mean, okay, it was easy. We compare just the minimizers and we got these energy estimates, but you could say, okay, why not? I mean, I'm not a fan of making variational schemes. I want to discretize my equation directly. So you could do that, of course, but then in order to get an a priori estimate, the standard way what you would do is to test by the velocity. This is something very standard. So in fact, if everything is discretized, we would test by this by this say discretized velocity. So the difference quotients. But if I do that, then I will be given terms which look like this one here. And actually in order to deduce an energy estimate, I would not like to have this gradients here or the derivatives here. I would actually like to estimate this by energies. But in order, I mean, this is just a chain rule. So if everything was continuous, the chain rule holds and everything is great. But once things are time discreet, there is a version of the discrete chain rule, which looks like this one and it would be correct. But it has a but this energy E has to be convex. And our energy is not convex and cannot be convex. So this is somehow the main reason why discretization of the equation directly will probably not lead to the right estimates. So this is something which tells you, no, you have to go to a variational approach. So this variational approach is really like the King's Road to get up to a reestimate in case of large deformations. OK, so once we have the existence of minimizers for our variational scheme, and once we have this energy estimate, we can derive an Euler Lagrange equation. In fact, the only reason why we can do this is that we had this regularized energy and we know that the Jacobian is bounded away from zero. Because in such a case, we can really take a variation and we can really arrive to an Euler Lagrange equation. We would not be able to do this without. But again, this is essential for us because convergence to the weak equation must be done on the level of these Euler Lagrange equations because we have the coupling of test functions. And this coupling of test functions is important for us because it gets risk for this difficult equality of tractions. So actually, once we have this Euler Lagrange equation and once we have the energy estimate, I just told you before, everything is ready and we can now really pass to the limit. The only maybe technical difficulties here to approximate test functions, right? Because the test functions depend on the solution, so they change in the convergence scheme. So one has to do a little some technical work there, but okay, so everything is more or less ready. So now you would say, okay, this was an easy situation because you skip the inertial term and the inertial terms are the only hard ones and you would be right. So now the question is, how can I extend this idea, which seems to be a good one, in case if inertia is present. So let me just have a look at here the equation for the solids. So now before we just consider this part here, the derivatives of the energy and this inertial term we just skipped. Okay, but now I want to get added back to the game, but now I have some incompatibility having here because we saw before that we need to have a variational approximation scheme in order to get a priori estimates because of this non-convexity. On the other hand, if I discretize this inertial term in the most, I mean by some midpoint scheme or whatever, then if I try to compare minimizers, this will never give me the right a priori estimate. Actually for this inertial term, the right thing to do would be to take the equation and to test the equation by the velocity as we did before. This is something I cannot do because I do not have the equations if I minimize. So you could say, oh, so maybe the Euler Lagrange equation and the minimizing problem are equivalent, but they are not because we are working with a non-convex energy. So somehow we have to come up with something and the idea here is that we introduce kind of two levels of approximation. We introduce one level of approximation for this inertial term and then I found a level of approximation for this energy terms. So let me just make a quick argument how this goes. So we take the first level. Oh, so then it's very quick because I only have like two more slides or something. So we first discretize this inertial term by some constant H and once this is discretized, we are in the same situation as we were in the parabolic case. Just like if the inertial term was even not there. So we use the same thing we did before. We may take some minimization problem, pass to the limit and then we are in the continuous case and can deduce further a priori estimates. So this is what I want to have. And let me just have let you make you a one second rough idea on what happens if I also add the inertial term for the fluid, which is the heart because we have the terms here. So what I have to do is I have to discretize it by something how the Navier-Stokes equation is actually deduced. I have to introduce kind of a flow map. So kind of Lagrangian coordinates for the fluid on a very short time step and discretize it by this flow map. Okay, so this is maybe only the rough idea. So let me just point you to the take home messages. We want to have this large structural deformations because of the non-convexity in the energy this calls for a variational approach. And if we want to include inertia and the variational approach, we need two levels of approximation. So thank you for your attention. And this is the reference. So thank you.
 I want to introduce a variation in approach to fluid structure interaction.  And this is work which was done in collaboration with Malte Kampschute and Sebastian Schwarzacher,  who are also both from Prague.  So let me just start what I want to talk about.  So what I want to talk about is fluid structure interaction.  So I mean, the world almost tells it everything.  So fluid structure interaction happens always when we have two materials.  One is kind of a fluid material and one is a solid material.  And they both move and have mutual movement somehow.  They influence each other on the movement.  So for example, I took this example here from some Journal of Computational Physics.  But of course, there are many examples and you can think of like a fluid which is flowing  next to an elastic membrane.  So this thing here is kind of an elastic membrane.  And as the fluid flows, of course, it exerts forces onto this elastic membrane and it starts  to move.  So in this example, which I just took from this Journal of Computational Physics, it  goes up and down and up and down.  So this elastic membrane moves quite a lot.  So what you can see already that there are many examples in which you have an interaction  between fluids and solids.  And we can also see that the solid is really moving and deforming a lot.  So what you can see is that the deformation of the solid can be large.  And it shouldn't be enough to say, OK, we think only of some small or rigid bodies because  really here you see an example where the deformation can be quite large.  OK, so that's what we want to consider.  So maybe that's the main thing of what will be the talk.  We will have a fluid structure interaction.  We will have some fluid and we have a solid which can really have large deformations.  So let me just go to the model and let me just tell you what the key ingredients of this  will be, which we have.  So we will, of course, fix everything to some fixed container.  So you can think of a fixed container like this.  So think here, this is like what happens at time zero or what I think of as a reference  configuration.  So this gray part here is my solid part, I think.  And the solid part is best described by its deformation, which I hear the note by Ita.  Because of course, we are mostly, for the solid, we are mostly interested where it is.  I mean, it's not so important how fast it moves.  It's really important where it is.  So therefore, we use this deformation here.  For the fluid, on the other hand, the main variable which describes what it does is not  really where it is because that's not so important for the fluid as it flows all the time.  But the main variable here is the velocity.  So that's also clear.  So what we have is a coupling between Lagrangian and Eulerian coordinates because of course  our solid is really described by this deformation, which you can see here.  It tells you where every point goes to and the solid by its velocity.  So once I have these key ingredients, which describe my materials, I can start up, set  up my modeling and for my modeling, I will just take the basic physical laws and those  should be satisfied.  So the basic physical laws are the balance of masses and momentum.  So we have the balance of mass and momentum for the fluid.  So that's of course, I mean, I could take a lot of possibilities for the balance of  mass and momentum, but I don't think about it a lot.  And I just take Navier-Stokes equation, which is maybe the easiest model I can think of  for a fluid and I take just the Newtonian fluid.  So I have here the inertial terms.  I have here the diffusion, which I will call dissipation.  I have some pressure and I have some forces and the divergence is zero because everything  is of course incompressible.  And for the solid, I have the balance of course also of mass, but this is not so important.  And I also have the balance of momentum, which here reads, okay, I have the inertial term.  I have some divergence of some pressure of some stress tensor sigma, which I will specify  in a minute, what is this stress tensor sigma.  So this is going to be like a modeling issue and they should be equal to the forces.  So it's also Newton's second law.  It's just the balance of momentum, nothing special here.  And of course, over the mutual boundary, which you hear, this is the mutual boundary between  the fluid and the solid, there should be some transition conditions.  These transition conditions tell you really how the influence between the two materials  goes on.  And in our case, we take maybe the most simple transition conditions we can think of.  We take the continuity of the velocity over this common boundary, which somehow in the  microscopical setting means that all the atoms move as the same fast.  And also, we take a continuity of traction, so it means the force over this boundary is  continuous.  So this here, the first here is really the continuity of velocity, which is clear, but  it's written down in this reference configuration because only there we can talk about the solid  velocity.  And the second here is, you already see, this is the stress tensor applied to the normal,  so this is the forces.  And here you just have the stress tensor stemming from the Navier-Stokes equation, revered  and again to the reference configuration.  So this is the boundary terms.  And now, okay, so you could say this is a complete model, but it's not because what  I'm missing here is really the stress tensor for the solid.  So the stress tensor for the solid needs to encode some modeling, and we will put a little  bit more complexity here into the solid or allow the stress tensor to be a little bit  more complex, particularly because I said that we want to have a look at these large  deformations.  It's okay.  We want to have large deformations.  So if you want to do that, then of course, the formation of the solid somehow specifies  the boundary of the main or the domain of definition for the fluid.  So of course, the deformation of the solid cannot be arbitrary.  In fact, in order to set everything up and be in the correct setting, the deformation  needs to be homomorphism at least.  So we need to choose our stress tensor rightly.  In fact, from the point of view of modeling solid materials, this is not like something  additional we ask for because when modeling solid materials, asking that formations to  be homomorphisms is something really standard or one should do this all the time.  So we will of course not consider all the materials.  So we will consider only such materials for which the stress tensor is given by two potentials.  So one of them is like an energy potential and one of them is like a dissipation potential.  So there are really many materials which fall into this class.  And I guess similar concept is known in fluid mechanics also, but in solid mechanics, this  is most often called the concept of generalized standard materials going back here to half  an a million.  And this is a concept which gives you like really a wide class of materials which you  can think of.  And for the talk here, I will further just think of a prototypical example for this energy  and dissipations which you want to consider.  So let me start with the energy here.  The prototypical energy I want to consider looks like this complicated expression here.  So let me just talk you a little bit through what I want to have.  So the first term here is just a norm.  So it tells you how far the gradient more or less is from the identity.  So it's an elastic term and here's like the tens of elastic constants.  So it's actually the standard Sanctum and Kyrchov energy.  So it really just tells me that my material actually would like to be relaxed and is really  in a stress-free state once there are no forces.  Once it's really an identity mapping here, the deformation gradient, so it's really  not the form at all.  We also consider some two additional terms.  So the first additional term I put here is like inverse of the determinant of the gradient  of the formation.  So the inverse of the Jacobian of the deformation.  And this is really a term which is needed and physical because once you start considering  large deformations, it never can be that you could compress a finite amount of the material  to zero volume with a finite energy.  This should never happen.  This should always be penalized.  So it's clear that any physically reasonable energy should blow up to infinity as the Jacobian  of the deformation goes to zero.  And here we just do it in this term.  And as a last thing, I consider a regularizing term.  So I add here the second gradient of the deformation to the energy, which is something  which is maybe not standard and which is sometimes called second grade materials.  And we do it for mathematical reasons mostly because we need more regularity.  Let me just tell you that if we have the second order term, we will know a little bit more  about the deformations of finite energy.  In particular, we will know that the Jacobian of any such deformation of finite energy will  be bounded away from zero.  So it will be positive.  There will exist some positive epsilon.  And we will know that this determinant will be always bigger than some epsilon.  This will tell you that this term here, the first term here, will actually not blow up  polynomially, but will be bounded, which is of course important.  But it will also be important for us from the point of view of many ways we will see  some examples in the talk later.  Let me just note that this energy overall is non-convex.  And it has to be non-convex because we added this Jacobian here.  And this is something physical.  So if we want to consider large deformations for the solid, we can never take a convex  energy and work with a convex energy.  Even this tankfrenam-Kirchhoff energy is non-convex, but OK, so this maybe could be convexified.  This is maybe not the point.  But the point here is that this blow up would never be possible.  OK.  So this is like the energy I have in mind.  And for the dissipation potential, I think of something very simple.  I take a quadratic dissipation here.  But of course, I have to take it quadratic, not in the deformation gradient, because that  would not be physical, because there would not be an independence of the observer.  But I have to do it in the Cauchy-Green-Penzo, which I do here.  But of course, the downside of it is that the dissipation potential starts to depend  on the state.  So we will see later why this is maybe something to highlight a little bit.  So OK.  So this is the modeling I want to consider.  So let me just wrap up and have a look at the strong formulation of what I want to do.  So I have the following system.  I have here my balance of momentum for the solid.  So I have here the inertial term.  I have here the derivative of the energy, the derivative of the dissipation potential,  the forces.  I have here the classical Navier-Stokes equation.  I have the divergence, which is somehow missing here.  The divergence of v equal to 0.  I have the transition conditions between the solid and the fluid.  And I have further initial and boundary conditions on this omega.  OK.  So this is the system I want to consider.  And what I ultimately want to do, I want to prove existence of weak solutions.  At best, we are a suitable time discretization scheme.  So why a time discretization scheme?  Because, of course, the domain changes all the time because the solid is moving.  So time discretization is somehow something which helps because we can fix the domain  for a very short time.  OK.  So this I want to do.  But I will just hit OK.  But I want, in order to make things simpler, I want to consider a simpler situation.  In fact, I want to, for most of the talk, I will, I will, I will, I will, I will, I will  consider a situation of slow movement.  So what does it mean, slow movement?  It means that the movement of the solid and the fluid is so slow that these inertial  terms, which are here, are not needed because they are so small, they can be neglected.  So this is something which I would call in the, in the, in the physics or in the modeling  of solid materials, we call all this quasi-static because it's almost static.  The solid is not moving a lot.  For the fluid, it's not really static because it's moving, but it's more, more steady flow.  So it's like quasi-static or quasi-steady situation.  And this is what I want to consider.  So these two inertial terms, I drop.  And this is the system that we will have a look at most of the talk.  OK.  So we have here a system which got parabolic, which consists now only of forces, which are  here on the right-hand side, the pressure.  But OK, so the pressure.  And derivatives of energy and dissipation potentials.  Exactly.  We have here like energy, we have here dissipation potential.  And this diffusion here in, in, in the fluid, I can also understand as a derivative of a  dissipation potential because of, yeah, it can be understood in the very same way because  it plays the same role.  OK.  So this is the strong system.  So let me now just highlight a little bit how the weak formulation of this system looks.  So if I want to design a weak formulation for this, of course, I have to take care of  these transition conditions.  And you would probably agree that the second transition condition here looks the most ugly.  So the continuity of tractions.  And it is even the most ugly because it features here the stress tensor, which is something  which is the least regular thing we know from, from here because, yeah.  So it would be best if we somehow could have this condition somehow disappear.  And this is what we can do in the weak formulation.  In fact, if we take our test functions, so let me just have a guide to your attention  at first to the very end of the weak formulation.  So if we take our test functions here, here phi is the test function in the solid and  psi is the overall test function, which lives on the whole of omega, but in particular is  also the test function for the fluid.  So if we take them coupled on the solid part in such a way that they depend on the actual  solution, then it's kind of a not too hard computation that in case everything is very  smooth for such test functions, this equality of tractions comes out automatically.  So let me just have a look.  So I have, I will call a pair at a weak solution of this parabolic fluid structure interaction  problem if it has some regularity, if they still satisfy the first coupling condition.  Okay, and if a weak formulation, which actually just consists of transferring here one derivative  and that's more or less this and tested by the right test function holds true.  So this is the weak formulation I want to consider.  And it's really just magical here that this equality of tractions need not to be considered  anymore.  Okay.  So for this weak formulation, I want to prove that from that there exists weak solutions  actually according to this week formulation.  So this is the theorem.  There exists a weak solution, more or less.  So let me just tell you how we prove this theorem.  So the proof of the theorem is based on some consecutive minimization in the spirit.  So we will set up a lot of minimization problems, which are time spent, the time stepping.  So sometimes this is called a time incremental minimization, and it's very much used in solid  mechanics.  So I just send you here and it actually goes back to the spirit of the Georgie or maybe  even earlier and is minimizing movements.  So let me have a look how this looks like.  So I want to set up this minimization schemes, which looks like minimization or minimizing  movements.  So I fix a time step size here.  I start with some initial data.  And if I assume that there is some that I have the eta tk given from my previous step,  then I construct the next time step as follows.  I minimize the energy plus the dissipation needed to go plus the dissipation somehow  needed to go from the last step to the step.  So I have here, I minimize here the energy.  I here take the dissipation potential, but I multiply this by tau.  I take here another dissipation potential, which comes from the fluid.  And I also add here the forces.  So this is what I minimize.  And I minimize it subject to eta just in a natural symbol of space, the divergence of  v should be equal to zero.  And there is a coupling condition on the boundary between which more or less corresponds to  this Dirichlet coupling condition, like the equality of velocities we had before.  So let me highlight some features of this scheme.  So first of all, there is no conditions on the stresses neither here.  So we didn't have it in the weak formulation.  And we also don't have it in this time stepping scheme, because the equality of stresses comes  out automatically from the minimization, because the energy plus dissipation is only  minimal if there is really no stress over the boundary and more or less.  Let me have a look that the scheme is somehow explicit implicit.  So it's implicit here in the energy.  This is important because it allows us to control the Jacobian and the geometry of the  whole thing.  It is implicit here in the v's and it's implicit in the dissipation in the somehow rate variable,  which is here the velocity and here also like the discretized velocity.  That is explicit in the state variables.  So the state variable would be here, the formation and also for the dissipation of the fluid,  the state variable would be the domain on which we live.  So in fact, during the minimization, all domains are fixed.  The domain for the fluid is given somehow from the last step, as well as here the same  holds here for the dissipation potential.  And this is actually needed in order to get the right Euler Lagrange equation for the  solid.  This is something which has already been used and for the fluid we needed to modify this  accordingly.  And also let me say that we have an explicit dependence here in the coupling condition.  So when we couple, we say that the velocities should be the same.  In order to couple here the velocities, you also have an explicit relation here.  And this is important to really get the stress balance now.  So once, okay, so now I can, I mean it's standard in the calculus of variations that I can prove  existence of minimizers for the scheme.  This is really something very, very simple.  So once I have existence of minimizers, I can start doing something with them.  And the advantage of having a variational scheme is that I can deduce an energy estimate  quite easily.  If I compare the energy of the minimizer to say the last step and zero, which is something  very like most natural thing one would come up with, I already have here an energy imbalance  which is the correct one.  So I have here that the energy of the current step plus the dissipation, this is all the  dissipation here, is bounded by the energy of the last step plus the force times.  So in order, so if I now estimate these force terms, which I can do say by, sorry, which  I can do by constant qualities here in order to estimate them into this dissipation, I  actually get an energy imbalance immediately.  So I have here that the energy plus the total integrated over time dissipation is bound  by some constant.  So this is like an immediate a priori estimate and actually this a priori estimate is more  or less everything that is needed to later to go to the to pass to the limit and to the  use the existence of weak solutions.  Okay, because we know that these energy and dissipations are coercive in the right way.  So this is actually already an a priori estimates, which tells you everything.  So let me at this point tell you why a variational scheme is so useful.  I mean, okay, it was easy.  We compare just the minimizers and we got these energy estimates, but you could say,  okay, why not?  I mean, I'm not a fan of making variational schemes.  I want to discretize my equation directly.  So you could do that, of course, but then in order to get an a priori estimate, the standard  way what you would do is to test by the velocity.  This is something very standard.  So in fact, if everything is discretized, we would test by this by this say discretized  velocity.  So the difference quotients.  But if I do that, then I will be given terms which look like this one here.  And actually in order to deduce an energy estimate, I would not like to have this gradients here  or the derivatives here.  I would actually like to estimate this by energies.  But in order, I mean, this is just a chain rule.  So if everything was continuous, the chain rule holds and everything is great.  But once things are time discreet, there is a version of the discrete chain rule, which  looks like this one and it would be correct.  But it has a but this energy E has to be convex.  And our energy is not convex and cannot be convex.  So this is somehow the main reason why discretization of the equation directly will probably not  lead to the right estimates.  So this is something which tells you, no, you have to go to a variational approach.  So this variational approach is really like the King's Road to get up to a reestimate  in case of large deformations.  OK, so once we have the existence of minimizers for our variational scheme, and once we have  this energy estimate, we can derive an Euler Lagrange equation.  In fact, the only reason why we can do this is that we had this regularized energy and  we know that the Jacobian is bounded away from zero.  Because in such a case, we can really take a variation and we can really arrive to an  Euler Lagrange equation.  We would not be able to do this without.  But again, this is essential for us because convergence to the weak equation must be done  on the level of these Euler Lagrange equations because we have the coupling of test functions.  And this coupling of test functions is important for us because it gets risk for this difficult  equality of tractions.  So actually, once we have this Euler Lagrange equation and once we have the energy estimate,  I just told you before, everything is ready and we can now really pass to the limit.  The only maybe technical difficulties here to approximate test functions, right?  Because the test functions depend on the solution, so they change in the convergence scheme.  So one has to do a little some technical work there, but okay, so everything is more or  less ready.  So now you would say, okay, this was an easy situation because you skip the inertial term  and the inertial terms are the only hard ones and you would be right.  So now the question is, how can I extend this idea, which seems to be a good one, in case  if inertia is present.  So let me just have a look at here the equation for the solids.  So now before we just consider this part here, the derivatives of the energy and this inertial  term we just skipped.  Okay, but now I want to get added back to the game, but now I have some incompatibility  having here because we saw before that we need to have a variational approximation scheme  in order to get a priori estimates because of this non-convexity.  On the other hand, if I discretize this inertial term in the most, I mean by some midpoint  scheme or whatever, then if I try to compare minimizers, this will never give me the right  a priori estimate.  Actually for this inertial term, the right thing to do would be to take the equation  and to test the equation by the velocity as we did before.  This is something I cannot do because I do not have the equations if I minimize.  So you could say, oh, so maybe the Euler Lagrange equation and the minimizing problem are equivalent,  but they are not because we are working with a non-convex energy.  So somehow we have to come up with something and the idea here is that we introduce kind  of two levels of approximation.  We introduce one level of approximation for this inertial term and then I found a level  of approximation for this energy terms.  So let me just make a quick argument how this goes.  So we take the first level.  Oh, so then it's very quick because I only have like two more slides or something.  So we first discretize this inertial term by some constant H and once this is discretized,  we are in the same situation as we were in the parabolic case.  Just like if the inertial term was even not there.  So we use the same thing we did before.  We may take some minimization problem, pass to the limit and then we are in the continuous  case and can deduce further a priori estimates.  So this is what I want to have.  And let me just have let you make you a one second rough idea on what happens if I also  add the inertial term for the fluid, which is the heart because we have the terms here.  So what I have to do is I have to discretize it by something how the Navier-Stokes equation  is actually deduced.  I have to introduce kind of a flow map.  So kind of Lagrangian coordinates for the fluid on a very short time step and discretize  it by this flow map.  Okay, so this is maybe only the rough idea.  So let me just point you to the take home messages.  We want to have this large structural deformations because of the non-convexity in the energy  this calls for a variational approach.  And if we want to include inertia and the variational approach, we need two levels of  approximation.  So thank you for your attention.  And this is the reference.  So thank you.
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I want to introduce a variation in approach to fluid structure interaction. And this is work which was done in collaboration with Malte Kampschute and Sebastian Schwarzacher, who are also both from Prague. So let me just start what I want to talk about. So what I want to talk about is fluid structure interaction. So I mean, the world almost tells it everything. So fluid structure interaction happens always when we have two materials. One is kind of a fluid material and one is a solid material. And they both move and have mutual movement somehow. They influence each other on the movement. So for example, I took this example here from some Journal of Computational Physics. But of course, there are many examples and you can think of like a fluid which is flowing next to an elastic membrane. So this thing here is kind of an elastic membrane. And as the fluid flows, of course, it exerts forces onto this elastic membrane and it starts to move. So in this example, which I just took from this Journal of Computational Physics, it goes up and down and up and down. So this elastic membrane moves quite a lot. So what you can see already that there are many examples in which you have an interaction between fluids and solids. And we can also see that the solid is really moving and deforming a lot. So what you can see is that the deformation of the solid can be large. And it shouldn't be enough to say, OK, we think only of some small or rigid bodies because really here you see an example where the deformation can be quite large. OK, so that's what we want to consider. So maybe that's the main thing of what will be the talk. We will have a fluid structure interaction. We will have some fluid and we have a solid which can really have large deformations. So let me just go to the model and let me just tell you what the key ingredients of this will be, which we have. So we will, of course, fix everything to some fixed container. So you can think of a fixed container like this. So think here, this is like what happens at time zero or what I think of as a reference configuration. So this gray part here is my solid part, I think. And the solid part is best described by its deformation, which I hear the note by Ita. Because of course, we are mostly, for the solid, we are mostly interested where it is. I mean, it's not so important how fast it moves. It's really important where it is. So therefore, we use this deformation here. For the fluid, on the other hand, the main variable which describes what it does is not really where it is because that's not so important for the fluid as it flows all the time. But the main variable here is the velocity. So that's also clear. So what we have is a coupling between Lagrangian and Eulerian coordinates because of course our solid is really described by this deformation, which you can see here. It tells you where every point goes to and the solid by its velocity. So once I have these key ingredients, which describe my materials, I can start up, set up my modeling and for my modeling, I will just take the basic physical laws and those should be satisfied. So the basic physical laws are the balance of masses and momentum. So we have the balance of mass and momentum for the fluid. So that's of course, I mean, I could take a lot of possibilities for the balance of mass and momentum, but I don't think about it a lot. And I just take Navier-Stokes equation, which is maybe the easiest model I can think of for a fluid and I take just the Newtonian fluid. So I have here the inertial terms. I have here the diffusion, which I will call dissipation. I have some pressure and I have some forces and the divergence is zero because everything is of course incompressible. And for the solid, I have the balance of course also of mass, but this is not so important. And I also have the balance of momentum, which here reads, okay, I have the inertial term. I have some divergence of some pressure of some stress tensor sigma, which I will specify in a minute, what is this stress tensor sigma. So this is going to be like a modeling issue and they should be equal to the forces. So it's also Newton's second law. It's just the balance of momentum, nothing special here. And of course, over the mutual boundary, which you hear, this is the mutual boundary between the fluid and the solid, there should be some transition conditions. These transition conditions tell you really how the influence between the two materials goes on. And in our case, we take maybe the most simple transition conditions we can think of. We take the continuity of the velocity over this common boundary, which somehow in the microscopical setting means that all the atoms move as the same fast. And also, we take a continuity of traction, so it means the force over this boundary is continuous. So this here, the first here is really the continuity of velocity, which is clear, but it's written down in this reference configuration because only there we can talk about the solid velocity. And the second here is, you already see, this is the stress tensor applied to the normal, so this is the forces. And here you just have the stress tensor stemming from the Navier-Stokes equation, revered and again to the reference configuration. So this is the boundary terms. And now, okay, so you could say this is a complete model, but it's not because what I'm missing here is really the stress tensor for the solid. So the stress tensor for the solid needs to encode some modeling, and we will put a little bit more complexity here into the solid or allow the stress tensor to be a little bit more complex, particularly because I said that we want to have a look at these large deformations. It's okay. We want to have large deformations. So if you want to do that, then of course, the formation of the solid somehow specifies the boundary of the main or the domain of definition for the fluid. So of course, the deformation of the solid cannot be arbitrary. In fact, in order to set everything up and be in the correct setting, the deformation needs to be homomorphism at least. So we need to choose our stress tensor rightly. In fact, from the point of view of modeling solid materials, this is not like something additional we ask for because when modeling solid materials, asking that formations to be homomorphisms is something really standard or one should do this all the time. So we will of course not consider all the materials. So we will consider only such materials for which the stress tensor is given by two potentials. So one of them is like an energy potential and one of them is like a dissipation potential. So there are really many materials which fall into this class. And I guess similar concept is known in fluid mechanics also, but in solid mechanics, this is most often called the concept of generalized standard materials going back here to half an a million. And this is a concept which gives you like really a wide class of materials which you can think of. And for the talk here, I will further just think of a prototypical example for this energy and dissipations which you want to consider. So let me start with the energy here. The prototypical energy I want to consider looks like this complicated expression here. So let me just talk you a little bit through what I want to have. So the first term here is just a norm. So it tells you how far the gradient more or less is from the identity. So it's an elastic term and here's like the tens of elastic constants. So it's actually the standard Sanctum and Kyrchov energy. So it really just tells me that my material actually would like to be relaxed and is really in a stress-free state once there are no forces. Once it's really an identity mapping here, the deformation gradient, so it's really not the form at all. We also consider some two additional terms. So the first additional term I put here is like inverse of the determinant of the gradient of the formation. So the inverse of the Jacobian of the deformation. And this is really a term which is needed and physical because once you start considering large deformations, it never can be that you could compress a finite amount of the material to zero volume with a finite energy. This should never happen. This should always be penalized. So it's clear that any physically reasonable energy should blow up to infinity as the Jacobian of the deformation goes to zero. And here we just do it in this term. And as a last thing, I consider a regularizing term. So I add here the second gradient of the deformation to the energy, which is something which is maybe not standard and which is sometimes called second grade materials. And we do it for mathematical reasons mostly because we need more regularity. Let me just tell you that if we have the second order term, we will know a little bit more about the deformations of finite energy. In particular, we will know that the Jacobian of any such deformation of finite energy will be bounded away from zero. So it will be positive. There will exist some positive epsilon. And we will know that this determinant will be always bigger than some epsilon. This will tell you that this term here, the first term here, will actually not blow up polynomially, but will be bounded, which is of course important. But it will also be important for us from the point of view of many ways we will see some examples in the talk later. Let me just note that this energy overall is non-convex. And it has to be non-convex because we added this Jacobian here. And this is something physical. So if we want to consider large deformations for the solid, we can never take a convex energy and work with a convex energy. Even this tankfrenam-Kirchhoff energy is non-convex, but OK, so this maybe could be convexified. This is maybe not the point. But the point here is that this blow up would never be possible. OK. So this is like the energy I have in mind. And for the dissipation potential, I think of something very simple. I take a quadratic dissipation here. But of course, I have to take it quadratic, not in the deformation gradient, because that would not be physical, because there would not be an independence of the observer. But I have to do it in the Cauchy-Green-Penzo, which I do here. But of course, the downside of it is that the dissipation potential starts to depend on the state. So we will see later why this is maybe something to highlight a little bit. So OK. So this is the modeling I want to consider. So let me just wrap up and have a look at the strong formulation of what I want to do. So I have the following system. I have here my balance of momentum for the solid. So I have here the inertial term. I have here the derivative of the energy, the derivative of the dissipation potential, the forces. I have here the classical Navier-Stokes equation. I have the divergence, which is somehow missing here. The divergence of v equal to 0. I have the transition conditions between the solid and the fluid. And I have further initial and boundary conditions on this omega. OK. So this is the system I want to consider. And what I ultimately want to do, I want to prove existence of weak solutions. At best, we are a suitable time discretization scheme. So why a time discretization scheme? Because, of course, the domain changes all the time because the solid is moving. So time discretization is somehow something which helps because we can fix the domain for a very short time. OK. So this I want to do. But I will just hit OK. But I want, in order to make things simpler, I want to consider a simpler situation. In fact, I want to, for most of the talk, I will, I will, I will, I will, I will, I will consider a situation of slow movement. So what does it mean, slow movement? It means that the movement of the solid and the fluid is so slow that these inertial terms, which are here, are not needed because they are so small, they can be neglected. So this is something which I would call in the, in the, in the physics or in the modeling of solid materials, we call all this quasi-static because it's almost static. The solid is not moving a lot. For the fluid, it's not really static because it's moving, but it's more, more steady flow. So it's like quasi-static or quasi-steady situation. And this is what I want to consider. So these two inertial terms, I drop. And this is the system that we will have a look at most of the talk. OK. So we have here a system which got parabolic, which consists now only of forces, which are here on the right-hand side, the pressure. But OK, so the pressure. And derivatives of energy and dissipation potentials. Exactly. We have here like energy, we have here dissipation potential. And this diffusion here in, in, in the fluid, I can also understand as a derivative of a dissipation potential because of, yeah, it can be understood in the very same way because it plays the same role. OK. So this is the strong system. So let me now just highlight a little bit how the weak formulation of this system looks. So if I want to design a weak formulation for this, of course, I have to take care of these transition conditions. And you would probably agree that the second transition condition here looks the most ugly. So the continuity of tractions. And it is even the most ugly because it features here the stress tensor, which is something which is the least regular thing we know from, from here because, yeah. So it would be best if we somehow could have this condition somehow disappear. And this is what we can do in the weak formulation. In fact, if we take our test functions, so let me just have a guide to your attention at first to the very end of the weak formulation. So if we take our test functions here, here phi is the test function in the solid and psi is the overall test function, which lives on the whole of omega, but in particular is also the test function for the fluid. So if we take them coupled on the solid part in such a way that they depend on the actual solution, then it's kind of a not too hard computation that in case everything is very smooth for such test functions, this equality of tractions comes out automatically. So let me just have a look. So I have, I will call a pair at a weak solution of this parabolic fluid structure interaction problem if it has some regularity, if they still satisfy the first coupling condition. Okay, and if a weak formulation, which actually just consists of transferring here one derivative and that's more or less this and tested by the right test function holds true. So this is the weak formulation I want to consider. And it's really just magical here that this equality of tractions need not to be considered anymore. Okay. So for this weak formulation, I want to prove that from that there exists weak solutions actually according to this week formulation. So this is the theorem. There exists a weak solution, more or less. So let me just tell you how we prove this theorem. So the proof of the theorem is based on some consecutive minimization in the spirit. So we will set up a lot of minimization problems, which are time spent, the time stepping. So sometimes this is called a time incremental minimization, and it's very much used in solid mechanics. So I just send you here and it actually goes back to the spirit of the Georgie or maybe even earlier and is minimizing movements. So let me have a look how this looks like. So I want to set up this minimization schemes, which looks like minimization or minimizing movements. So I fix a time step size here. I start with some initial data. And if I assume that there is some that I have the eta tk given from my previous step, then I construct the next time step as follows. I minimize the energy plus the dissipation needed to go plus the dissipation somehow needed to go from the last step to the step. So I have here, I minimize here the energy. I here take the dissipation potential, but I multiply this by tau. I take here another dissipation potential, which comes from the fluid. And I also add here the forces. So this is what I minimize. And I minimize it subject to eta just in a natural symbol of space, the divergence of v should be equal to zero. And there is a coupling condition on the boundary between which more or less corresponds to this Dirichlet coupling condition, like the equality of velocities we had before. So let me highlight some features of this scheme. So first of all, there is no conditions on the stresses neither here. So we didn't have it in the weak formulation. And we also don't have it in this time stepping scheme, because the equality of stresses comes out automatically from the minimization, because the energy plus dissipation is only minimal if there is really no stress over the boundary and more or less. Let me have a look that the scheme is somehow explicit implicit. So it's implicit here in the energy. This is important because it allows us to control the Jacobian and the geometry of the whole thing. It is implicit here in the v's and it's implicit in the dissipation in the somehow rate variable, which is here the velocity and here also like the discretized velocity. That is explicit in the state variables. So the state variable would be here, the formation and also for the dissipation of the fluid, the state variable would be the domain on which we live. So in fact, during the minimization, all domains are fixed. The domain for the fluid is given somehow from the last step, as well as here the same holds here for the dissipation potential. And this is actually needed in order to get the right Euler Lagrange equation for the solid. This is something which has already been used and for the fluid we needed to modify this accordingly. And also let me say that we have an explicit dependence here in the coupling condition. So when we couple, we say that the velocities should be the same. In order to couple here the velocities, you also have an explicit relation here. And this is important to really get the stress balance now. So once, okay, so now I can, I mean it's standard in the calculus of variations that I can prove existence of minimizers for the scheme. This is really something very, very simple. So once I have existence of minimizers, I can start doing something with them. And the advantage of having a variational scheme is that I can deduce an energy estimate quite easily. If I compare the energy of the minimizer to say the last step and zero, which is something very like most natural thing one would come up with, I already have here an energy imbalance which is the correct one. So I have here that the energy of the current step plus the dissipation, this is all the dissipation here, is bounded by the energy of the last step plus the force times. So in order, so if I now estimate these force terms, which I can do say by, sorry, which I can do by constant qualities here in order to estimate them into this dissipation, I actually get an energy imbalance immediately. So I have here that the energy plus the total integrated over time dissipation is bound by some constant. So this is like an immediate a priori estimate and actually this a priori estimate is more or less everything that is needed to later to go to the to pass to the limit and to the use the existence of weak solutions. Okay, because we know that these energy and dissipations are coercive in the right way. So this is actually already an a priori estimates, which tells you everything. So let me at this point tell you why a variational scheme is so useful. I mean, okay, it was easy. We compare just the minimizers and we got these energy estimates, but you could say, okay, why not? I mean, I'm not a fan of making variational schemes. I want to discretize my equation directly. So you could do that, of course, but then in order to get an a priori estimate, the standard way what you would do is to test by the velocity. This is something very standard. So in fact, if everything is discretized, we would test by this by this say discretized velocity. So the difference quotients. But if I do that, then I will be given terms which look like this one here. And actually in order to deduce an energy estimate, I would not like to have this gradients here or the derivatives here. I would actually like to estimate this by energies. But in order, I mean, this is just a chain rule. So if everything was continuous, the chain rule holds and everything is great. But once things are time discreet, there is a version of the discrete chain rule, which looks like this one and it would be correct. But it has a but this energy E has to be convex. And our energy is not convex and cannot be convex. So this is somehow the main reason why discretization of the equation directly will probably not lead to the right estimates. So this is something which tells you, no, you have to go to a variational approach. So this variational approach is really like the King's Road to get up to a reestimate in case of large deformations. OK, so once we have the existence of minimizers for our variational scheme, and once we have this energy estimate, we can derive an Euler Lagrange equation. In fact, the only reason why we can do this is that we had this regularized energy and we know that the Jacobian is bounded away from zero. Because in such a case, we can really take a variation and we can really arrive to an Euler Lagrange equation. We would not be able to do this without. But again, this is essential for us because convergence to the weak equation must be done on the level of these Euler Lagrange equations because we have the coupling of test functions. And this coupling of test functions is important for us because it gets risk for this difficult equality of tractions. So actually, once we have this Euler Lagrange equation and once we have the energy estimate, I just told you before, everything is ready and we can now really pass to the limit. The only maybe technical difficulties here to approximate test functions, right? Because the test functions depend on the solution, so they change in the convergence scheme. So one has to do a little some technical work there, but okay, so everything is more or less ready. So now you would say, okay, this was an easy situation because you skip the inertial term and the inertial terms are the only hard ones and you would be right. So now the question is, how can I extend this idea, which seems to be a good one, in case if inertia is present. So let me just have a look at here the equation for the solids. So now before we just consider this part here, the derivatives of the energy and this inertial term we just skipped. Okay, but now I want to get added back to the game, but now I have some incompatibility having here because we saw before that we need to have a variational approximation scheme in order to get a priori estimates because of this non-convexity. On the other hand, if I discretize this inertial term in the most, I mean by some midpoint scheme or whatever, then if I try to compare minimizers, this will never give me the right a priori estimate. Actually for this inertial term, the right thing to do would be to take the equation and to test the equation by the velocity as we did before. This is something I cannot do because I do not have the equations if I minimize. So you could say, oh, so maybe the Euler Lagrange equation and the minimizing problem are equivalent, but they are not because we are working with a non-convex energy. So somehow we have to come up with something and the idea here is that we introduce kind of two levels of approximation. We introduce one level of approximation for this inertial term and then I found a level of approximation for this energy terms. So let me just make a quick argument how this goes. So we take the first level. Oh, so then it's very quick because I only have like two more slides or something. So we first discretize this inertial term by some constant H and once this is discretized, we are in the same situation as we were in the parabolic case. Just like if the inertial term was even not there. So we use the same thing we did before. We may take some minimization problem, pass to the limit and then we are in the continuous case and can deduce further a priori estimates. So this is what I want to have. And let me just have let you make you a one second rough idea on what happens if I also add the inertial term for the fluid, which is the heart because we have the terms here. So what I have to do is I have to discretize it by something how the Navier-Stokes equation is actually deduced. I have to introduce kind of a flow map. So kind of Lagrangian coordinates for the fluid on a very short time step and discretize it by this flow map. Okay, so this is maybe only the rough idea. So let me just point you to the take home messages. We want to have this large structural deformations because of the non-convexity in the energy this calls for a variational approach. And if we want to include inertia and the variational approach, we need two levels of approximation. So thank you for your attention. And this is the reference. So thank you.
some transition conditions. These transition also, but in solid mechanics, this is most often if we want to consider large deformations for the with a convex energy. Even this want to prove existence of weak solutions. At In fact, I want to, for most of the talk, I will, interaction problem if it has some regularity, if it actually goes back to the spirit of the there is no conditions on the stresses neither important because it allows us to control the approximation scheme in order to get a priori this is discretized, we are in the same situation inertia and the variational approach, we need two