 We are getting into the forte of the summer school. It's clearly that it's getting more and more difficult. And like we said already in the beginning, it's like what we try to do is build up the program, going from the pathophysiology, the image acquisition, and further on to the modeling, and then towards the applications. And one of the things which is also important is the numerical methods and everything which is associated with that. Because in the end, when we want to do modeling, this is very, very important. And it's my great pleasure to have Eugenio Ognate here today, who is one of the gurus in numerical modeling. And he's working at the UPC, he's professor at UPC here in Barcelona, and he's the founding father of SIMNE, the International Center for Numerical Methods in Engineering. And already it's now, you said 30 years ago, that SIMNE started, and already 30 years ago he had the vision that simulations is something which is extremely important in engineering. And not only in biomedicine like we do, but obviously it's a technology which is much, much older, but it's very important for having somebody that has been supporting this kind of technology for years and years. So it's an honor to have you here today and show us a little bit of what you have been doing. Okay, well, thank you for this kind of words and thank you for all being here. It's not 7 a.m., but anyhow, maybe it's early for some of the participants in the course. This is a work that mostly is related to the topic of this seminar. But I would like to give you a broadened overview of how we approach the solution of problems of interest probably to you in the biomedicine and biomechanics, but that have many other applications in engineering and applied science. And they all have a particular feature. They combine fluids, solids, and particles, okay? So you can think fluid as blood, solid as any component that has strength, and particles as any object of a small or large dimension. And this is the team. Dr. Souda is here in the audience and Professor Edelson and some other members of the team in Thimne. Now, okay, I tried to go slowly and skip as many details, but not all the details, because after all, you want to put your hands on codes and program, not only the pilots of users of codes only. So I'm going to tell you how we model solids and particles using combination of finite element and discrete methods. And you can read here, we can formulate, actually we can set up a unified approach for handling these complex problems, involving fluids, solids, and particles, and with applications to a broad number of problems in engineering and biomedicine. Okay, just very, given the extent of the lecture, although already half an hour has almost gone, I would like to recall why we are doing all this, because we are going to understand what's going on around us. And we are going to use these tools that will give us solutions in terms of numbers. In fact, it's interesting that what we are trying to model is reality with the help of a mathematical model. And here we start with the first problem, what is reality? And here we will see that each of us probably has a different sense of what is reality. And for that problem, for that reason, any solution that we obtain is already influenced by our vision of reality. For instance, there are many realities, as you know, in the observation of the world, we can go from the subatomic scale to the great numbers in continuing for modeling components at the emitter scale. So for each of these realities, we need appropriate numerical methods. So this is the first thing to learn is which is our scale of interest? What do you have to decide? And then what are you looking for in that scale of interest? This is another important decision because you can get many results, but might not be what you're looking for or you might be using the wrong method for the scale that you are looking at. But we are interested in problems in mechanics dealing with objects that deform, like cathedrals or nuclear power plants or historical buildings, or looking to the flow of water or air in problems of interest to the automotive industry, to the ship building industry, et cetera, or manufacturing processes. Still, I don't put any bio application here. And all these problems share the same commonality that they are all looking for answers to questions like how big, how much, when the structure will fail, how much noise are we getting and how efficient is the manufacturing method. And for that, I think this is where we use numerical methods. We are going to combine technical knowledge. We need to know what we are talking about and I'm not going to talk too much about things that I don't know today, like for instance, biomedicine. I try to talk about things that I know more. And then you have to use knowledge of mathematics and computer science to quantify the solution to these problems. Also, this is a very important sentence. We have to be a modest and not really be tied up to any numerical method. We are all the time trying to really marry bring together methods because we know that all, that's not such a perfect numerical method. This is an statement from George Box, which is an economist. He was referring to models in economy, but I think I took this sentence and actually I adapted. So the usefulness of the method is what counts if they are useful. And also this is 30 years after Thymnia was created, this is how we present our work. We are only part of a system where we actually worry about a node here in the system, which is the predictive node, computations. And the two of the nodes is the devices, experiments and knowledge data. And this is with these three systems together will help us to get a better product or to solve a particular problem. You can see that they are all connected. They are all connected. So society don't worry too much about each of these independent nodes. They want to solve a problem. And this is particularly relevant in biomechanics and biomedicine. And you know that we can model systems very simply. We've seen simplified models like networks and this is done to model the blood flow in the human body or structures or networks of gas or electricity. This is what we call the discrete system. But in general we are worried about what we will call continuing systems. This is an example of a continuing system and we're going to use discretized method and you are all familiar with the concept of discretization. It's going to the infinite numbers of degrees of freedom of the problem to a finite number of degrees of freedom through a discretization. And the most common method is finite elements but this is one of the many methods, finite volumes, finite differences. You can use many methods. And in fact this method is not new. It was used by a chemist, the concept to compute the ratio between the length of the circumference to the diameter, as you see, by discretizing the circumference. And you can see that by using polygons of a larger number of nodes, he was able to quantify an unknown, an unknown which is pi and using upper and lower bounds depending on if it is since subscribing the polygon. So this is the essence of what we're doing with the discretization method and it had to pass something like 600 years until someone else actually brought from China extended the concept of discretization to areas and he was able to bound pi using the same concept of dividing the continuum, dividing the unknown domain, excuse me, dividing the domain where we're going to compute the unknown in sub domains where we can compute areas or to some lengths. So now we know what we're going to do. We're going to discretize our body of interest and this is applied extensively in all the problems that we solve every day, in structures, in the manufacturing industry, what we see and what actually, believe me, what our society, let's say, our customers they don't want to see is meshes. But we are using these meshes behind the objects of interest. You can see representation of meshes here. And I bring here this example because we are next to the ACBAC tower and I brought this case of analysis of how we use finite elements to discretize the tower and to model the formation of the tower and it's this particular tower, I don't know if you know that it is all sustained by the walls, by the external walls, okay? And what we are looking is the stresses in the walls and inside is only the core for the elevators and the floors, okay? So this is an example of a very, I will say, very interesting building just 200 meters from this room. And the same has been analyzed and this is, I use the word pathology all the time to model how structures will behave until they fail, until they die, I use these concepts. So we take a structure like a cathedral and for instance here you will see that we actually increase the load until it fails and then it collapses. So this concept of a structure of pathology has many similarities to what we do when we want to analyze the strength of bones or processes, et cetera, okay? So numerical methods are used here to see how structures behave in the, I will say healthy status when there are no damage, like here, or to see how much they will live. For instance, now we are studying the life of the three nuclear power plants in Catalonia, asco one, asco two, and Valdejo's. The question is, how much life do they have so they can have or not permission to function? Okay, so here's a heart, this is the first. And everything that I have said, this is some marks chapel in Venice. Everything I have said for this variety of structures applies when we want to see how will this human, this component behaves. But you see, this goes beyond our imagination. Lancôme actually advertises this product in terms of the good features it has on the skin. And they say that they use finite elements to model the behavior of this cream on the skin. You see, so this is not so much new. This is almost 10 years old, okay? But I thought it was appropriate to bring it here. So okay, we have here reality, we have our models. We have our computational method. Again, each reality will have a model, a structure, and then we have results. And what we need is to make sure that we know how this behaves. And this is how we need invariably, and this will always be like that. We need a reference from experiments. And the experiments link directly with reality. So when we compare our results with experiments from reality, then we know that our model and our perception of reality is good. But if we use analytical solutions, analytical solutions to compare our results, these analytical solutions are already polluted by our model. Okay, so this is an important concept. It's the difference between validation and verification. So you said reality I model with a model with a particular differential equation. I obtain a solution and I compare, maybe our differential equation is wrong. If I compare with experiments, then I know that my model and the method is correct. I thought I brought this here, because this is very relevant. And this is very subtle. But ask any of your colleagues if they know the difference between verification and validation, and this is not so well known. But how about other methods that are really increasingly popular? There is a conference that we run. There is a journal that we run on particular methods. And I would like to bring this because after all this lecture goes about particles. So there are methods that are based on modeling a continuum as a collection of particles, or they model a collection of particles used by the individual particles. So these are perceived as discretization methods or to model a real collection of particles. And these are very simple because using only the interactions through the contacts and applying Newton law, some forces equal mass time acceleration or torque equal inertia times the angular acceleration. In integrating these equations, I will come back to this fast later. And they can model a variety of problems. This is only some examples. And you will be surprised of the number of applications in pharmacy, in mining industry, in food industry, in chemical engineering, in civil engineering, in biomechanics that involve particles like this and methods like this. For instance, these were in the previous example, these were non-cohesive particles. And his is so in model as a collection of particles which is excavated. So this is another application of interest. But then we are talking about fluids in particles. So we are going to put all these particles in a fluid. And we need to know what we are doing with fluids because again, what do we mean by fluids? It's air, it's water, it's bentonite. So here I would like to just to tell you that we have methods that deal with flows within cavities. This will be blood flow. This is what we call internal flows or flow within this room. And flows with the free surface. There is a big difference between each of these methods. And we could talk about Eulerian and Lagrangian method. They are more appropriate for one or each other application. For instance, I can see that for airplanes what we have this concept of the virtual wind tunnel also for cars and for telescopes, this will be what we will call internal flows because we are going to put the object in a case and we're going to analyze how the air will flow through this. However, we have a landslide or a ship cruising on waters. Then this is what we will call external flows because we are going... So here we have what we would like to call the virtual wind tunnel which is a domain where we will put the car and we will let the air flow through the tunnel. We will need to actually manipulate the geometry. This is complex and to actually generate the mesh and we have to clean all the details in the car. And then we have to put the car in the domain and we set a mesh which is fine enough to capture all the details around the car. And this is not a fine mesh but we could not afford a finer mesh. This was done some years ago by a master's student from Imperial College. And this is the objective here is to compute nice pictures like these ones but after that what we want to compute is the drag, the resistance of the car in the wind. So I just wanted to bring the attention to this concept of the virtual wind tunnel which is in the line of what we will call virtual labs. You have a virtual lab to see how a structure will deform. A virtual wind tunnel to see how a car or an airplane will flow. And we have also the concept of the virtual towing tank. What is a towing tank? A towing tank is a domain like this. Again, it's a domain. I introduce in this concept of virtual lab now. It's a domain when you put your model of your actual ship, scale one over 40 and you tow it and see the resistance. And then you have to assume that there will be waves in the sea, okay? And this is an animation of how a mesh is generated with the ship inside. This is our virtual towing tank. There is one in Madrid, one kilometer long. You have to tow it. The longest one is in Washington, in the David Taylor. You see the mesh is created around an object. This is an American Cup ship, okay? This is only a conceptual example to see, do you understand what is a virtual wind tunnel or virtual. Okay, so this is an American Cup. I don't want to stop here, but you can see the details of how the mesh is refined next to the body. I will go further, faster, I don't want to, and this is supposed to go on, yes. Visualization is very important. Please translate all these cases into your problem of interest in biomechanics. I will reach biomechanics, but before that, we had to really master this, and Moa will say practical problems in engineering. And so, okay, so, and but don't forget after that, what we want is numbers. We want to compute the wave elevation to compute the drag force. We want to compute the drag force, the resistance of the force. The number of interest is, what is the resistance of the car in the air? What is the resistance of the shipping waves? And if you fail to capture this wavey solution, which is very difficult to obtain, the numbers that you will get will be useless, with nice colors, nice pictures, but a scale of interest, the ship, number of interest, the resistance. Let's go into more complex problems, and like this, tsunami flows, or flows that have waves, or flows moving in a kind of a turbulence with a free surface, and then put particles there. And what is a particle? A particle is a colloid, a particle is a cow, or a car. So we are going to talk about this very small particles, a small and large. Okay, you could say micro, macro and large, but micro and macro, very small and small and large. And this is an example of a tremendous particulate flow. Here you have cars, and even ships flowing very large, with very small particles. Here you have particles which are large, and particles contained within these containers, which is like a mineral ore or grain. You can have here 250 tons blocks that are wiped away during the night by waves, so they get large particles. And here we have the latest project that we are carrying. We have particles of ice, that a model with discrete elements in a liquid, which is the sea, excuse me, which is the sea, and they have buoyancy. Many problems of interest in the Americas, but here you see this could be an atheroma plaque, or this could be a human, I don't remember the name, I have blood, it's called, it will come back later. That is eroded by the effect of the fluid. This is very typical in civil engineering. You have a dam, you have over-topping flow, and the Y and the dam, this happened in Valencia in 1982. This disappears and this creates a tremendous collapse. I will put examples of this problem related to the blood flow later. In excavation problems there are many problems of this type, and here this is not an artery, this is a six kilometers pipe of 20 centimeters diameter, where inside we have bentonite, which is kind of non-Newtonian fluid, transporting the particles of that drill at the end. So this is a problem that we are solving, and we are actually using this technology to model blood flow, of course. Here is blood flow, and here is blood flow, and finally, very simple problem of blood flow, transporting particles inside. Transporting particles, yes, finally? So let's talk a little bit about how we do this. Right, I have seen there is a lecture later on Eulerian and La Ranche, and I don't know if you are familiar with this, but this is actually, it's a test for what we know about fluid mechanics. Let's not go further if we don't understand this clearly. When we talk about Eulerian description, is when we are looking to the river, we have a fixed domain, and the flow passes, okay, passes to the domain. When we talk about La Ranche, I am a La Ranche object. I travel with, and I carry my coordinate system. Particles are always La Ranche, okay, because we follow the particles. The fluid could be Eulerian or La Ranche. This is clear, but the particles will be always La Ranche. Okay, you could have a combination of both methods, but we don't talk about that. So we have methods, in fact, the difference in mathematics is that when you have a fixed control volume, you have, in the definition of the acceleration, you have the convective derivative, you use the velocity. And this term is nonlinear, you see? Velocity times gradient of velocity. And this is the term that is very difficult to model in fluid mechanics. Using Eulerian description. You have Eulerian description. You put particles, okay, we'll see in a minute, and you have problems with this term. In La Ranche's description, you don't have this term. It's simpler, but you need to update the domain. The flow particles will move. You have to follow them. Okay, this is very important. Can we write a code that deals with the two problems at the same time? Yes, we can. We have done it. And this is what we call unify, Eulerian La Ranche formulation. Now, another terminology that is extremely confusing. Believe me, we run a conference. This is the seventh edition on particle methods. The next one is in Hanover. And here people come and talk about particle methods. What is a particle method? Okay. Particles are real. Are colloids, or are sand, or are blood particles? Well, or not. Well, many methods are called particle methods, and in fact, they are used to discretize a continuum. So they don't use real particles. For instance, in these methods, you have methods in which particles they don't have mass. SPH, you're going to talk here about SPH later on, but this discretization method that uses no meshes, but the concept of particles in P, a smooth particle dynamic, is to discretize a continuum. And even the particles can have mass, like in DM or the material point method. But the objective here is not to analyze physical particles, but to discretize a continuum. For instance, you've discretized a domain to analyze fluid flow using a particle method. Otherwise, the particle method, like the discrete element method, can be used to model physical particles, granular matter or individual portions of a continuum. So the DM here is in the two sides. You can use it to discretize a continuum, like a concrete block, or you can use it to model millions of particles flowing in the fluid. Okay. This is confusing, very confusing. For instance, the particle faradelman method, it is the method that we use to solve fluid. On top of that, we put the discrete element method that we use to model the particles. Here are the Lagrangian particles, you see, in red, plus a Lagrangian solid. Confusing? No. You have objects here that will move in the fluid, they will fall down, and we are also going to track the fluid nodes in blue, fluid nodes that are going to move. And inside this mesh, we are going to have millions of particles that are going to move with the fluid. This is what we use to model the tsunami, for instance. You will see examples. This is what we call Lagrangian particulate fluid. It's a Lagrangian fluid with particles linked to a solid, and solids are always Lagrangian. Solid particles are always Lagrangian. In structural mechanics, we never talk about Lagrangian. They don't even know what I'm talking about. It's only when you talk to the fluid community that you really start the debate. Okay? I'll skip this. Embedded formulation. I'll skip this. This is too technical. This is interesting, but... Now let's talk about discrete element method, because you can make your living only with this method, not even knowing about fluids or solids. I have seen these communities. Why is it very appealing? Because it's very simple. It's very simple and very effective. We have a code that you can use, actually. It's very effective. For the mining industry, for instance, here you have particles and they move. And what is what you need to get an accurate solution? Well, what you need is to model the contact between the... This is a project that we have with the company called Metallogenia. They want to see the efficiency of these loading machines and the wear in these cutting tools. Well, these are particles. This is used in pharmacy. There are many applications. I will skip these. These mixers. These applications in pharmacy, all the applications in pharmacy involve discrete element methods. For instance, the fracture of a pill. Nice. Or the coating of particles. You put particles. You mix them. We don't have time to see all these. You put them around. Then you can have a fluid inside. But at the same time, you might have a cohesive part of, like an atheroma plaque. You want to say, I want to break this cohesive solid, like this specimen, load it until it fails, and you can use DM to model that. DM used here as a discretization method. And what is the only thing that matters in the DM? To model the interaction between a particle in the neighbors. An interaction which is in the normal direction. And you have a spring plus a damper. And in the tangential direction, you have a spring plus a damper. That's all. That's all. Well, but you have to calibrate these parameters. So the accuracy of a discrete element method comes in the calibration of these parameters. So you have forces in the normal and tangential directions. You have to relate these forces to the stresses. So you have very simple equations and mechanics. This is a strength of materials of the first year. And you reach a question like that force in the normal direction related to the displacements and the velocities of the relative velocities between the interacting particles. And you have coefficients that depend on the elasticity modulus of the actual material. So this is the value that you obtain in experiments. And with it you obtain the microparameters. But how about that you might ask, but this is for cohesive particles. Yes, this is true. If the particles are non-cohesive, you only need to worry about the tangential direction. Tangential directions. And you obtain the relationship between the tangential stresses and the forces and the displacements. Which is the friction. Right. Failure. You can have failure criteria. And because, after all, what you want is to break this pill. You load it. You obtain force until it breaks. And you want to see maybe the strength of this pill. A spinning. Or it's a concrete or cement. And for that you need to really account for fracture, for failure loads. Very simple failure loads. There are papers published on this. This is ice, for instance. Ice. You take an ice block. You load it. You break the cohesion between the particles. And it breaks. And then there are experiments that give you the failure load. And we compute. We compare. So I recall we have a method that models discrete particles. Each of them individually or in groups. And the friction between them. This is for non-cohesive material, like grain, ballast, or many other applications. And we have cohesive particles that also, that have cohesive loads that break. And they are useful to, for instance, this is an example of how we will break or we will test. This is for the food industry. These particles will break. Interesting. This is another problem. You have particles falling. Rock. On a net. And this net is also modeled with particles. Interesting. This is a curiosity. Now we know how to model particles, cohesive or non-cohesive. We know how to break them. Now we are going to put them in the fluid. We put them in the fluid. So if the particles are very large, like if I model myself with, and fall it into water, I might use finite elements to model the large particles. If the particles are very, not smaller, I use the Discretelman method. For instance, this is cereals within a cup of milk. And this is always strongly criticized in the UK, because they always tell me that nobody puts the milk and then the cereals, so they're all the way around. But you can see here how nice it looks. And the particles are modeled at the end and the fluid, the milk with the depth. So these large particles move following Newton law. I already talked about it. So all we need is to compute the forces of the fluid on the particles. And we have here interacting domains. And then we will move the particles. This is the particle falling in a tube. It could be a vessel. But here you have, you see a problem. Do we fall, do we, the mesh will follow the deformation? Nice results are always guaranteed. Analytical results exist for this problem. So you can see that moving the mesh is very complex. It's complex. But it's part of the problem when we want to model, we will have high fidelity modeling of a large particle moving in one. Let's talk about this unified formulation. Perhaps I will only tell you, I put some questions, but that you can model Eulerian and Lagrangian with a single mathematical setting. This is what we are talking about. But I will skip that. And perhaps this is, it was new to us. Now it's very familiar. Perhaps it's not so much well known for you. You know the questions of fluid mechanics. Momentum equation, you see. This is the generalized form Eulerian with the convective terms, Lagrangian without the convective term. So acceleration, the sum of forces, the forces due to the internal stresses plus the body forces. So these are the standard equations. This equals zero is the standard equations in mechanics, in fluids and solids. Sum of forces equals zero. Sum of forces including the acceleration equals zero. Now, if you put particles inside the fluid, we use what we call the embedded approach. The particles are put inside the fluid. They occupy a certain space. And they reduce the fraction of available fluid here. You see, if this is zero, that means that this is all occupied by particles. If this is one, this means there are no particles. And the effect is put here in the fluid equation. So this is the force particle to fluid that is perceived by the fluid due to the particles embedded. So this is the general concept. You put the particles, the particles occupy a space and they have an effect on the momentum equation. Great. Well, this is so simple, yes, it's so simple. And the same in the mass balance because you have less fluid in the differential domain here. You have less fluid is reduced by the particles. So you have to account this in the mass conservation equation. And this is also in red, the terms that appear when there are particles inside the fluid. Good. If there are no particles, we have the standard equations, momentum and mass balance. Great. This is the compressibility of the fluid. So now we know how to include the particles. Now let's look to the particles. We know that the fluid see the particles. In the particles see the fluid because in the forces that govern the motion of the particles we have the weight, we have the contact forces and this force here that comes from this term. So now we know how to model particles within the fluid. The fluid will see the effect of the particles and the particle will see the effect of the fluid. Some details, but very simple. It's a repetition. Mass time acceleration equals forces. What are the forces that the particles see? The weight? The contact in the normal and the tangential direction. And the fluid forces, which is the archimedes force, the buoyancy force, the archimedes, plus the drag. And the drag is very difficult to compute analytically. Excuse me, the drag is very difficult to compute because you will have to do real fidelity simulation to compute the boundary layers, blah, blah, blah. But there are many analytical solutions for simple shapes in fluids. So we took drag forces from the aerodynamic literature and we applied this analytical drag to the particle. Great. So now we can give this to any of you or any of our students to program because now we know all the details and the details. We don't need to understand now, but please make sure that all the details are well documented otherwise you cannot actually go further. Right. I will skip this because, and I will skip this. After that, you need to put this in the differential equations. You have to apply, for instance, in finite elements, you have to have a variational form for the momentum and for the mass volume equation and then you have to discretize. This is before discretization. And then we use always, in this case, a mixed formulation. What does it mean a mixed formulation? Our variables are velocities and pressure in the fluid. Hey, also in the solid. This formulation includes Lagrangian. How about if this domain includes a solid and not a fluid? It's the same. So we can model a solid falling in the fluid. The solid is Lagrangian and the fluid is whatever we want. A Lagrangian. So this is not only a general formulation for fluids and particles. It's for a continuum in particles. This is very nice. We like to unify. We dream, it's like Pythagoras. He wanted to govern the world with numbers. We are mobodes, which will not attempt that. But we want a unified formulation. There is a penalty, a little penalty. In solids, you have an elastic solid. You use displacements as variables. Here we use velocities and pressure. But do we need pressures? No, you don't need pressure because you can eliminate pressure in a solid. But we use pressures and velocities. So we use the same variables, velocities and pressures in the fluid and in the solid. There is a thesis, Alessandro Franci, that recently has finished, which is, all this is explained in detail. And the equations, this is momentum equation. This is the acceleration terms. This is the term including the viscosities and the pressure term. And this is the mass conservation. Where we have here the particle forces. And here also the particle forces. This is all for a, this is the end of the equation. It's good. I think you got the message a little bit. So we have, we move the particles. We solve for the fluid and the solid. And then for Lagrangian flows, we move the mesh. So let me tell you which kind of Lagrangian flow we use. Because this is very powerful. We believe more powerful than any other method. And there are no equations after this. It's called particle finite element method. PFM. And this is explained in a graphical picture. Let's look at this as a cloud of particles. And here particles refer to material points. They are used to discretize. So this blue will be water or air. This red will be another fluid. Or it will be a solid, maybe a ship. This cloud have some fixed points. So this is what we call cloud time n. Then we press a button. This is computational geometry tools. You can download it from the web. This is used in design, in design of clothes or design of cars. You can recover the domain, the boundaries of the domain. There is a method called alpha shape. They will give you this. Of course the quality of this boundary will depend on the number of particles. But cloud to domain. To solve a differential equation, you need a domain of analysis because you need boundary conditions. Don't forget also the famous story that God created the differential equations and the devil created the boundary conditions. So the boundary conditions, for that you need a domain to apply these devil curves conditions. And then I like finite volumes. I like finite differences. I like maybe SPH. Any method you use to solve your differential equations here. I love finite elements. No, I don't because we have 35 years of experience of finite elements. So we press a button and then we generate the mesh. And then this mesh inherits the properties of these points here. You can see here we have a flying particle here and a flying subdomain. Then we solve the equations. We move the nodes because we are a grandian. We move, you see? We have moved here. We have moved. And then we repeat. We move and we throw the mesh to the basket. We throw it. We don't want the mesh. Here it is. Then we go back. Cloud domain, domain mesh, mesh cloud. Cloud domain, domain mesh, mesh cloud. So simple. Yes. For instance, a body that is eroded inside the human body under forces due to fluid. Cloud domain, domain mesh. After some time, maybe this is a polymer. This is rock cutting. I could have put here another name. You see how this method gives me an extremely powerful to do things like this. This was done almost 15 years ago. Maybe no, 11, 2004. And this is what? Nobody wants to see the mesh. The mesh. This is slow in this case. But anyway, this is better. You see? Very difficult problem. And then we don't show to people the mesh. And this is the method. At each time step, we have two new duties that in final elements or in normal numerical methods you don't have. We need to redefine the domain at each time step and we need to redefine the mesh. 20 years ago, remeshing at each time step, it was a burden. Now it is not. Because they are very fast mesh generators. You can regenerate in parallel. And you get a lot of freedom. A lot of freedom. Let me put this example. This was also an academic example. You see? There are experiments for this case. And you see here. This is very nice. This node, you see these runs? Automatically, it is captured by the wall. The method, because it has this geometrical tool, prevents the nodes from leaving the domain. So it's extremely powerful. Also, allow us to see the contact of the fluid with boundaries and also the contact between objects through this green interface. Excuse me. How? You have a solid falling or a fluid, like in the previous case, approaching the boundary. And through the mesh, we identify that we are close to the boundary. And here we apply to these elements some contact forces in the vertical and tangential direction. And we can model contact in this manner. And you say, oh, is this accurate? Well, as much as you want to pay, as much as you want to pay for this contact interface, this is not very accurate. Enough, because what we are interested in is to see how these tetrapods will fall into water. So these are all ingredients, so very powerful ingredients to solve problems. A la carte. Let's see what problems we would like to solve. Okay, academic problems. And let's see how we are doing. Okay. This is another case. There are many, many academic examples that one can solve. You can do experiments yourself. You can see the, you know, compare... Lagrangian fluid, Lagrangian solid individual particle. This is the case. How about many particles? Well, for that, first we start with one. Okay. And we throw a particle embedded in the fluid. Let me go back. Particles now embedded. Are you already Spain? The fluid see the particles? The particles see the fluid? Yes. This is the case. I will go further. We have the velocity. We have an experimental value for that. And we have here a collection of particles falling. You see? Embedded. Great. 3D. And we start playing. Not playing. Applying. This is air bubbles. Interacting with physical particles. And some particles are dragged, you see, by the air to the top. This is the famous example in chemical engineering. A fluid-sized bed. You see? Particles flowing. But these are very dense collection of particles. Still they are embedded. Still they are embedded. So here maybe the fluid coefficient ratio that I explained. Maybe it's close to 1. Maybe 0.8. But it still is embedded. It works. And this is a very interesting part. You see how this is very interesting for the human application. You will see. You see water, blood hitting the surface and eroding the surface. And then transporting. When the fluid forces exceed a certain amount, this is used from abrasive wear. Then this part of the domain is transported inside the fluid. A jet hitting here. You see? It's creating particles. Now we are creating particles. Transporting the particles. And they fall here, you see? How they fall. And then they accumulate. Interesting. Many applications. This is the Valencia case. The mini toes, we call it. All these particles have been dragged. We don't collect them. Then here, this is another case. Fluid passing. And here these particles have been dragged. So now we can drag particles. We can erode collection of particles with cohesion. Okay. So we can do many problems. And I show you a thinking of your personal problem that you would like to solve. Oh, I have to skip this. This is a large particle. A lorry. Lagrangian fluid. The question is 20 tons lorry in a real way. You see? It's very dangerous to drive when you are next to a storm next to the harbor. This is a submarine landslide. We have a project now in this case. This is a wave. This is a wave. This is an academic case. Oh, yes. Sorry. There is a wave here coming. Eroding. When the friction exceeds a certain value, you will see. And this is a lorry, which is there quietly standing. You see this is eroding. And then after some time, it falls down. And particles here will accumulate. Creating particles and affecting what is around. Well, the landslide, I skip that one. This is the Lituya landslide. They're falling into the reservoir, creating a wave. Let me talk about this. This is the tsunami. This was an advanced ground project. Small and large particles together being mixed. In this case, they are combined. So this is when we are being trained. Now we master or we can handle very small particles together with large particles. And still we will not break this wall. We would like to break it. We are going to break it. We have a project with our colleagues in Japan. Of course, they're investing a lot of money in this. And hopefully, with these methods, we will solve cases of their interest. Oh, this is a validation case that we did to see the effect of the particles. Too slow, anyway. I skip this. This is of interest. Dredging, but this has many applications also in medicine. Use suction from a cohesive part and take it to another. This is the vessel problem that I mentioned to you. We would like to erode and transport these particles. Transport these particles. I skip this. And this is an example, let's say, of a jet on a cohesive soil that will break the soil and will transport the particles. And the particles are collected here. This is a very simple test that I have been used to see if the method works. And, of course, these particles are transported in different fluids. This is 3D, of course. There are different fluids. If the fluid is water, the fluid goes much faster than the particles. You see, when the fluid goes to 0.25 meters per second, the particles go very slow, less than 0.02. But if the fluid is black, or let's say bentonite, in this case, mud, the particles and the fluid go to the same speed. That's why dense fluids are used to transport particles in the oil industry. Very good conclusion. Okay, I'm almost finishing with these applications of your interest, of course. More close to your interests. But now we can model problems like this one. This one is particles falling, hitting the tip of the drill. Particles flowing in a multi-phase fluid. We will see an application of this in the vascular device in a minute. You see, very high fidelity simulations. This is the PFM. This is also two fluids, particles flowing. This is what it's called oil recovery. You push water inside the soil to push the oil out of the soil. This is the case here. You have an oil mass, and as water flows, this oil is recovered. This oil will go to an adjacent soil. This is called... And here you see, Eulerian fluid. We know what it means. Yes, the fluid is Eulerian, because this is a fixed domain. So we don't abandon Eulerian, not at all. If there is not a free surface, we use Eulerian fluid. Finally, I have 10, 12 slides where you will see similar applications of a head-to-head plane into the cardiovascular sector and the upper airways sector. This is some of the problems where this method can be applied. This is more familiar to you. Some of these are our results, and some of these I have borrowed from other colleagues. This is the human airways, and this is the case you probably know. And I have to tell you that this is a very simple case from what we have seen. So we inject particles according to the breathing process, and we see the properties of the particles that are retained or not. I will cover more on the next example. This I think we have solved. Can you inject particles? And some particles are retained due to some friction conditions. Again, this is in the upper airways. So here is the effect of breathing. And this is very important. You can see for aerosol design, anything to do with tobacco and other applications. I'm sure I would like to comment, because you are more experts. Eduardo here can comment. This is the similar case. I think one of those we have solved in our code and the other we didn't. This is not our result. This is in Karlsruhe. It's interesting what they say here. They talk about the human nose. It's a complex organ, blah, blah, blah. So the objective of this study was to learn about how and where particles spread and deposit in the human nasal cavity. This is the objective. Fine. So now we have methods that can be applied to this problem a little bit further. So this is the geometrical aspects. And at the end, I would have to say that not relatively simple applications of flow, air flow, again with particles, and they check where the particles deposit. And I cannot comment on the outcome, because this is not something that we have done, but I just wanted to bring you this direct application of the method. Tracers in a nasal channel. Same problem. This is all related to air flow with particles. They check the velocities. This is what? Now we go to blood flow. Blood flow is much more common. Here particles are large. They're embedded again. We have embedded the particles. And they check different aspects related to the properties of blood flow. I'm not an expert in blood flow, so I just wanted to give you... Interesting to see here, the small and large particles traveling together in that application. Let me go to another case here. And these are deformable particles. We could deform them as well, yes. Embedded. They are all using the embedded approach. I mean, the particles are on top of the fluid mesh. The mesh here is Eulerian, I'm sure. If it is closed cavity, mesh is most of the time Eulerian, because there's no free surface. And then we follow this trend of applying the forces to the fluid and to the particles. Another case here. Well, you see? Hybrid method. This is interesting because hybrid methods, like we have used finite elements, PFM and particles, they use lattice Boltzmann and finite elements. I don't have time to comment on that particular mixture of methods. That is similar to what we're doing. And they check on different values of the hematocrit. They get the... This is the good one, and they disregard the other. So they use these simulation tools to get the results of interest. I come back to the same concept. Scale of interest and results of interest. So this is what you get. Another case here. These applications you can borrow. I'm sure you can have this presentation. I will leave with the organizers. This is ventricular flow. So this particular case, and I don't know what was the objective of this study, but combine, again, the deformable wall of the ventricle with modeling of particles inside. Another case of... This is, again, going to the upper aways. So this is a very simple conceptual geometry. Particles moving up and down. This is interesting. This is what I mentioned about particles, trying to make progress through these lattice and some paths and some others don't pass. Any of these problems we could solve. I'll just put them here as an example. This is interesting of a plaquetas, platelet aggregation. Here we'll have the accumulation of red. In fact, red blood cells. The plaquettes are also red here. At the end they get this agglomeration of formation, in this case of this cohesive platelet. Another case here, particles. The diameter of the particles is color. The bigger in red, the smaller one in blue. And this is another study for whatever purpose. Well, time is passing. This case we have solving theme. We have a clot here. We are going to have flat flowing and we are going to use the same erosion method that I have explained for the riverbed to model how this is eroding. You see? And then it travels for whatever purpose. Interesting. I think you can see the interest of this clearly. I will not make any more comments. Another case. This is another clot here. We have fluid passing. We don't plot the fluid. And then it moves. Some of it is fully desiccated but some goes in blocks. Okay. This is an example for my commercial vendor. Particles on a valve. There are not so many actually. I haven't found any more. And I look for many. So the conclusion. Don't read it. I will just tell you in words. You can model the fluid with a Lagrangian or a Eulerian frame. You use a Lagrangian frame. The Lagrangian frame for the fluid is more adequate when you have free surfaces. So maybe in some applications in biomechanics. Otherwise on top of the Eulerian or Lagrangian fluid you have to put particles, the Lagrangian, or solids, the Lagrangian too. Okay. So that's the conclusion. The method that we use to model the fluid is the particle fan element method. And all this is in a free and open source code that you are welcome to use it. Okay. This is our repository of computation and knowledge in theme there. Okay. So all this there is an open source. An open source means that you can actually look at it. It's hard, you know. It's like the British library is an open source that you have to study. Or the Chinese... Chinese... I'm sure that many of you can do it. And there are many details here. For instance, we have all the fluid, structure, and them solvers, and many other solvers. This is used to store the knowledge so that you will add incremental knowledge on that. But we start from the knowledge. Also for the structure and interaction. And then there are many applications. I think I don't want to bother you with this. There are many applications about Python interfaces. I'll put it at the end of the lecture, just in case I had to leave it with you. This is how... And there are many applications including solids, fluids. This is the embedded applications that I didn't comment also for... Okay. I think this is the end. There's the particles there as well. And many more. So this is the end. I invite you to look at it. If you are interested in learning more details, if you want just to pilot the code, you can also use it. And I wish you good luck in your career. Thank you very much. Thank you very much for this impressive talk with many, many, many nice movies. I think a lot of us are jealous that you can really produce these things. So to start with some of the questions, it's like we're all doing simulations in biomedicine, and we know that it's still a very, very evolving domain. So how, with your experience in the industrial world, how mature is it in the industry and how many places is it really used as a day-to-day tool? Well, I can tell you for what I know. I know three or four groups in the world that are very active in this field. The group of professional learners in George Mason University, they use it for the design of devices, particularly with aneurysm. It has probably the best collection of aneurysm simulated. Also, Professor Charles Taylor from Stanford, he has a company, the company is called? Heart Flow. Heart Flow as well. I know he was one of the beginners in these fields, and he had made progress. Also, for valve design, the group in the University of Aachen is very active. But this is what I know. Probably what I don't know is bigger than that. But if you talk about the non-medical applications, how much is it used there actually? For the non-medical applications, all the examples you have seen, they have been supported by industry, all of them. Is it something that they use on a daily basis, or is it something, like you say, as an academic exercise? No, no, no, not academic. Not academic. The oil industry, before they actually designed it well, they tried to do simulation of how the drill will break the soil. It is used for design purposes. In the manufacturing sector, before you manufacture a product that has to be the form, it's simulated because the manufacturing process is very expensive. So the virtual lab is almost a reality in the manufacturing sector. Sheet foaming, casting, welding, additive manufacturing, they all use extensive simulation. So I think the biomedical sector is a bit behind, but it's coming. Particularly for device, for device design, how this device will behave, the questions, what will happen if, also for prosthesis design. So I guess that we are converging to make these tools usable. And you showed some examples of failures of buildings and things like that. How reliable are currently these simulations? I always tell my students that nobody's going to pay you for anything that they know. So basically, we get many questions and requests for problems that are sensitive, and there's no answer, you know. How long will this nuclear power live with this criteria? What is the death criteria? That the steel cables will lose 25% of the strength. So we tell them 10 more years or 25 more years. They have to believe it. But this is the same risk, you know, if you are an astronaut, you take risk. So I think that, and there is a big, I think this is the big beauty of these methods, they are predictive. So what we do is we calibrate on problems that we know the answer, complex problems, we break things. We break this, this is not expensive to break. And then we will really bet that the life that we are going to predict for this nuclear reactor is the good one. We are analyzing now the Fukushima plant, the corium inside the plant. Corium is the metal that has been melt, the refrigerator, the pipes that have been melted. And they are polluted with nuclear. So there's no way to go there in many years. So they are doing simulation to guess where the corium pits are inside. So they can send a robot inside. Anyway, very interesting. Any questions? I already took your time. Hi, Jerome. Oh, Jerome. Okay. So thank you, Renio, for this great talk. I came a bit late, unfortunately. So maybe some of my questions were already answered in your first slides. So I have three questions. The first question I have is that when you start to apply particle methods to biological systems, you're confronted to the great difficulty of defining the laws that will rule the interaction of the particles among each other and with the boundaries. And the common physical laws that are using in civil engineering or in mechanical engineering are basically failing. And so from your experience, can you comment on whether you already had to cope with this problem in civil engineering or whether you were aware about people who could successfully validate models of particle interactions for biological applications? This is the great challenge. I already mentioned at the beginning that models are simple, but the interacting forces is the key. And these forces are different for every material. They have to be calibrated for any material. Different from cement for clay, soil. And these refer both to the accumulation then to the desegregation. So this is a challenge for every material that you want to model with particle soils. And I'm not surprised that the civil engineering laws will fail because they are not two laws that are applied to each material. So the only answer is that we need extensive calibration of the methods with the experiments. I cannot give any recommendation because still we are fighting to calibrate these methods for three different materials, which is rock, cement, which is paste basically, and concrete, and some clays. I think that perhaps the human body will be more like in soil mechanics, more softer materials. So there is a future there ahead. And so following on that, during the discussions, can this solver admit changes along time of particle interaction rules, depending for example on fluid velocities, on instantaneous fluid viscosity, and so on? Or would it generate conservation problems? No, I don't think so. I think, well, I don't like to say that there will not be no problems, but the laws, the friction laws could be, you could define as a function of time. I don't see any mechanical problem in that. In fact, sometimes they are activated when the particles hit the walls. If not, they are switched off. So the friction laws could be really fine or activated during the transient solution. Yes, they can. I'll talk to my last questions if I can. So if you take an extreme case of Lagrangian solide and Lagrangian Eulerian fluid, so with let's say 20% of volume of particles and cohesive particles, are you able to simulate a pro-elastic phenomena like soil consolidation or for some thinking about tissue consolidation? I think so. This is a target now. This is a target to model porous material with fluid inside. Using these methods and also with the formable particles, the formable particles. But this line is very new. This is all very new. The particle world is mature, not mature, but it's very popular in communities like chemical engineering, for instance. It's amazing. Or in mining, for the mining industry, chemical engineering is the root of embedded methods in particle history. But now, pharmacy, food, and many other industrial companies or sectors are applying the particle methods. In soil mechanics and in civil engineering, it's relatively new. These are being finished these days on the topic. This marriage of continuum fluid and particles is also new. For instance, we have had 150 PhD thesis and only three or four particle methods in the last three or four years, so it's new. We will see many more of that in the future. Also, because you need to have a tool that can really make you see the life of the mechanics of the three fields involved. And you see in our community, at least in civil engineering, people used to work in vertical lines. So in mechanics, they have their own codes, and they don't master the fluid part. So now we need these transverse themes. But it's going in this direction, also for funding. Thank you so much for the very nice talk. Sometimes when we try to build models in biomedicine and specifically models from imaging data, what happens is that we don't have all the data that the model needs for boundary condition, but also sometimes we have much more data that the model perhaps doesn't need. An example is, say we want to model in triventricular flow, and we actually have measurements of velocity inside the domain, like from an image, for example, of velocities. So when I work with modelers, they normally say, no, no, but just give me the inlet and the outlet. Then I do the model. So if I want to actually give more data, they just cannot put it into the model. And I was wondering whether do you think that there is a way how this modeling paradigms can accommodate for this sparsely localized data, not necessarily at the boundaries, and if perhaps it's the particle framework that will allow to just give properties to particles inside the domain? It will. It will. It will show you some inconsistency somehow. If there is an error in the inlet or the outlet, you could actually use the inside information to predict the inlet and the outlet. And see if this really corresponds to what you expected. So there is never too much information. You are very lucky to have all that information. How to use it? Well, yes, to your advantage. But this information will be very useful. What is difficult will be too much to tell the inlet, the outlet, and the inside information consistent. But this is the target. So you probably can define an optimization problem where you have to minimize the error between the inlet, the outlet, and the inside field. This is an interesting problem. Thanks. If you have like a pure, solid mechanics problem, do you see any advantages of particles, or is it just... Yes, when you have multi-fracture, for instance, in explosions. Let's say you have a... Well, in explosions, for instance, in Cibula engineering, we use explosions in rock mechanics, in for tunneling. But let's say you have a concrete block. This is for security. Well, security is now a part of the HP 2020. So you want to see the effect of an explosion for an accident or human induce, it's called, on the wall of the aircraft structure. So you have initially a solid, which is a concrete block, and you put inside a charge, explodes, generates 2,000 particles of different size, and these particles really interact with the structure. So this is particle to solid interaction. These are the kinds of problems. Industrial explosion is for the construction industry, tunneling or queries, canteras, and for security, for security and defense. Any other questions? Sorry, I took all the time of the... Well, I gained the time, I recovered particle time. Absolutely perfectly in time. Thank you very much again.