 Very energetic good afternoon to all the dignitaries and friends present here. My name is Gauri Verma and I work as a scientific officer with Baba Atomic Search Centre. I would like to thank all the organizers of Icon India, specially Ritesh, Vikrant for helping out set up this particular, I had some technical issues and they were able to help me out in this. I would like to thank them a lot. So today I am presenting my topic which is the infiniteness of finite elements explained to Phoenix. First of all, I would like to start my presentation with this famous quote by a very famous computational scientist from Massachusetts Institute of Technology. He said, So here we will talk about what is finite elements and what is infinite about it. What basically is the, what infiniteness here, what we mean by that. So this is the outline of my presentation. I will be talking about the infiniteness of finite elements. It is exploration through the Phoenix package. We will be doing some case studies here. First case study will be on the solution of the heat equation using the Phoenix package. Secondly, the second case study would be the solution for the equation of linear elasticity and then finally there will be a summary of what we have presented and what we have talked about here. As you can see, this is what do you mean by infiniteness? Infiniteness in finite elements. As you can see, this is what do you mean by infiniteness? Infiniteness in finite elements basically means its infinite applications. As you can see on this page, there are four applications I have shown of finite elements. Have you ever wondered basically how when car manufacturers basically when they design a car, they build a car and they, when they, before basically selling it to the market people, how they basically arrive at such a beautiful optimum design. They basically use this particular tool called finite element analysis. This particular tool, finite element analysis is based on a finite element method. It is a mathematical tool which is, which have been used by softwares to use for physical problems basically. As you can see here, I am showing you four applications. First of all, in the, on the left corner, you can see a car having the doing the, having the solutions of displacement and with having a crash analysis. For the top right, you can see it's a pressure vessel. Basically tan upon which when you apply some external loads, external pressures, how it deforms. These colors, variation of colors, they basically tell about the regions of criticality. As you can see, the red color shows that region is both critical and the blue color which is on the top side. As you can see on the right side and right side of the region. It means it's of less criticality. Now on the left, you can see electromagnetic, it's a transformer basically and using finite element model, they're trying to basically estimate what are the temperatures one can achieve in this transformer when it's working. And on my bottom right, you can see the model of a heart. It's a finite element model of the heart. What is happening here is, the heart is pumping blood. So they have exactly modeled how a human heart is and using the full structure interaction module, they are basically showing what is basically happening here. How the heart pumps the blood. So this finite element basically, it has a name word. It is a word called finite in its naming, but it has infinite applications. It is used almost in every practical engineering application. Even people from science background use it. So it may have a name, finite elements, but its applications are infinite. So basically, when I will talk about what is finite element method, what basically is finite element method. So mathematical interpretation, it should take, it is a numerical technique for finding approximate solutions to boundary value problems for partial differential equations. Basically what it means is that to have a physical system, you have to know what solutions you can achieve. For that, you have to make a mathematical model and then you have to solve that mathematical model using some mathematical tools like matrices and all those stuff. And then you have to arrive at a solution. And physically, what does, it has, what physical interpretation it has? Basically, the content. Basically what physical interpretation it gives is the continuous physical model is divided into multiple finite pieces called the elements and the laws of nature are applied on the generic element. The results are then recombined to represent the continuum. Basically what happens, if you have a 3D structure, let us assume we have a book. It is a 3D book. It is a rectangular, it is basically a rectangular cuboid. You can see, it is basically any book you take. It is a big volume. It can be broken into multiple small, small, small, small volumes, 3D volumes. And those 3D volumes can be divided into multiple, as you can see, we are divided into multiple 3D volumes. You can, if you want to analyze, like if you put a load on that particular book, like you put a weight of your, like a paper weight you put on that book. How that load of the paper weight will have an effect on the book? The book I am taking just as an example. How will that have an effect? This can be computed using finite elements. So what we do here, I will show you the next slide. Basically what finite element method is basically what we are talking about. Basically what is the relationship between finite element method and finite element analysis? In simple terms, finite element method is a method for dividing up a very complicated problem into small elements that can be solved in relation to each other. It is applied to those problems which either have complicated geometries, loadings and material properties where analytical solutions are difficult to achieve. For example, as I told you earlier, we have taken a book and we are putting a paper weight on it. As you are putting a paper weight on it, what is happening is it is getting pressed at some location where the paper weight is in contact with the book. So now that you see this book is a very simple structure. It is basically a rectangle extruded. So it is a kind of a cuboid. But if you have a very difficult geometry like a guard where you have a lot of contours and a lot of protrusions and extrusions and all those stuff, then using basic mathematical laws and physics techniques which we have learned in school and colleges may not be applicable. And if you want to get very good results, like more accurate, have to go towards accuracy, then we have to use some mathematical tools to achieve those. And those mathematical tool has to be solved using computers. And calculations may give you just few numbers, but you have to be more accurate when you are going for a very, something very important, something very costly, something very practical. So that particular thing is achieved using finite elements. As you can see, finite element analysis is the application of finite element method for performing numerical simulations. Now I will take some... As you can see, how the finite element analysis works, I am talking about this here. You see, you have a physical system. Physical system like as I told you about a book having a weight on it, or example a car moving at a speed of 60 kmph on the road. This car is basically going to see some track forces because of the air, which is passing over it, passing through it basically. So you have a typical system. It has to be modeled mathematically in order to use mathematical tools. It has to have a simple mathematical representation. How will you idealize the whole physical continuum into a mathematical model? That needs to be done. Then after that, that mathematical model is further discretized into more simpler elements using finite element methods. And then you find solution for those discrete models using the finite element tools and other numerical techniques to find the solution for those. And once you see if this is okay, then it's fine. And otherwise if you find that there are some more tweaking required, then you again remodel the whole car thing with more intricacies and you come out with a solution. And it's an iterative process until and unless you are very much satisfied and you have a gut feeling that now this design will work and it's okay and you have done a lot of tests on it also. So as to validate whatever you have done is correct or not. Then you have achieved your solution. So how we have gone through? We have taken a physical system which is basically a car. Let's take a car. We have modeled that car physically on a computer. Let's take a computer and we have modeled that car. Then we take the whole model and divide that particular model into small, small elements. And then what we do? Using those, you solve for each element, you solve whatever the loadings are on that element. And then you get a solution for it. And then what you do? Ultimately also all the elements, you combine those solutions and you get the solution for the whole car. So this is how the finite element basically works. Now how does finite elements, as I spoke earlier, how it was basically in finite element analysis, the continuum or the physical system or the model is represented as an assembly of smaller subdivisions called finite elements. And together they are called mesh. As you can see on the right, it is a small geometry which I have divided into number of small pieces. These are basically rectangular pieces as you can see. These are called meshes. Each rectangular piece is called an element and together they are called a mesh. These elements are interconnected at nodes, at the points where they are connected. Those are called nodes. And basically we find solutions for those node values. The nodes usually lie on the element boundaries and are shared with adjacent or the neighboring connected elements. The mesh is programmed to contain the material and structural properties which define the behavior of the structure under certain loading conditions. As I told you, when you make the model of the car, the car could be made up of some kind of steel, some steel alloy would be. So you have to put that material property of that steel on that model so that when you apply a loading, it has an effect of that particular material. You see, because otherwise you don't put any material, you won't get any other piece. It's just like putting something, some loading on something which has no property. That should not happen. You have to put the material model. What kind of material it is? Is it a steel? Is it a aluminum? Is it something else? Is it whatever it is? So you have to put a material model on that and you put some loading. Loading could be a force, or could be a pressure, or could be some kind of a displacement. Placement means some kind of a deniation you put. So that has to be put on it and then you are able to get a solution for that. So now we will say how this finite element works, how it works basically. Now these nodes are distributed with certain density throughout the model depending on the anticipated resolution level on a particular area. The mesh is programmed to contain the material and structured properties which define the behavior of the structure under certain loading conditions. The variation of the field variable, it could be temperature, displacement, pressure, velocity etc in the model is defined, determined by approximating the variation of the field variables inside an element by a simple interpolation function also called the basis function. Basically what it is saying here is that what you want to determine in that particular car model if you take and you have put some material on it and you are applying some loading. What do you want to determine in that thing? It is like you do want to determine what the displacements are going to be like what deformation it can take. For example if you take a car model and you are travelling on a 60 km per hour speed and hit a barrier, you hit something like a crash test basically what we do. So what do you want to determine? Do you want to determine how much energy is absorbed, how much deformation is happening on the car or something? That is your field variable, the primal variable you call it and that you have to determine, that is what is mentioned here. And since you have put, you have taken appropriate mathematical model to model particular distinct. You have taken the correct material properties, you have applied correct loadings, basically the pressure load of the force load. You get the correct value of energy absorbed by the car or the displacement that the car is facing or seeing or whatever. You can achieve that using finite elements. The differential equation of any order cannot be solved. That is very difficult and particularly for very complex geometries it is very difficult to do that. So what do you do? You simplify that physical model. Sorry, if you simplify that differential equation. What you do is basically you simplify it bringing the order down like if you have a second order differential equation you bring the order to the first order. So that it becomes easy for you. That lowering of the order, that is what is called the weak form. The original equation which you have is called the strong form and when you load the order it is called the weak form. Now what happens? You use this weak form for finite element methods as the basic governing equation for which you are going to solve the solution for which you are going to get the results. So that is the weak form that I am talking about. And how do you solve it? You apply a lot of mathematical tools like you make matrices and you find out the stiffness matrix and all those stuff. There are quite mathematical terms. So what you want to happen is you form these mathematics equations and all those defined for each node. As I told you that each element at its boundary have multiple nodes connected to it. So you find those values at each node and you try to find what are the solutions like displacement, energy and whatever you want is the value on that particular node that you try to find out. Next. So I will tell you next thing. How this finite element analysis helps designers, basically the new age designers who are coming after doing some mechanical engineering and civil engineering after they are basically engineering courses undergraduates or anyone in first part, people from R and D of a car company or like in my particular organization like Pawan Tabaksa Center where we make nuclear reactors basically. So how that helps? So what happens is this finite element in health that it makes our life very simple. It somewhat replaces the need for an extensive and expensive physical experiments. Because for each thing you cannot do an experiment. Like if you want to model like for a car you do crash test analysis. That is okay. But for like a rocket, how much experiment can you do? Cannot really fly a real size rocket and crash it somewhere and see what can happen to the material of it. So there you do a scale down test. That's okay. And parallelly those tests are compared with the results of the finite elements and seen how these results are. Secondly, it is easily applied to complex irregular shaped objects composed of different materials and having complex boundary conditions. Like if you have very complex geometry like very difficult contours hand solved problems and calculations of calculators it will be very difficult to get accurate results. You might get somewhat closer results with 15 to 20% accuracy that could be there. But if you want a very accurate result you might have to you may not be possible using pen and paper. You might have to use some medical technique. And secondly, it is applicable to both linear and non-linear problems. What do you mean by that? Like problems where input and output behave linearly. Like whatever given input even output is linearly proportional to it that is called a linear problem. For example, you can take any example like particularly if on the book the earlier problems we have taken we have put a paper weight on it the more heavy weight you apply on the book the more deformation the book will see a more depression kind of thing the book will see. That is the linearity. Non-linear problems are difficult and different where the relation is not linear For example, those problems are highly non-linear and this finite element analysis is applied to a wide variety of problems like solid mechanics, fluid mechanics, electromagnetic, biomechanics, heat transfer etc. So it has really a wide application Now we will talk about Phoenix software Phoenix package which we adopted First of all, it is an open source package It is a python based software which basically solves the partial differential equations using Python and C++ libraries Basically we have a lot of libraries made for this and we basically if you want to do some calculation or we do a particular type of problem then we particularly import a particular module a particular library and then using those libraries whatever tools are there in those libraries we try to find out the solution and then if you talk about Phoenix it is an automated programming environment for differential equations Basically what Phoenix does is I will explain in the further slides how it helps us Phoenix provides a high level user interface which allows easy reproduction of mathematical formulation and rapid implementation of high performance solvers So it's that Then Phoenix was basically the Phoenix was created in 2003 under Phoenix research and software project with similar research laboratory University of Cambridge University of Chicago Technical University APH Royal Institute of Technology and some more institutes Okay, I got it The slides are not changing but I will try to wait I will re-thodecast myself So that We were expecting some glitches sometimes We are all connected from home So that's all right Yeah, I hope so So Basically what's happening is this Phoenix platform was created Basically it was a project taken up in 2003 by some laboratories and technical universities around the world and they together developed it and it's an open source package It's freely available in the internet and it has literally wide and very powerful and it has a lot of it solves a lot of problems physical problems physical and engineering problems So this is the architecture of the Phoenix application You can see on the left hand side you have this Phoenix application in the yellow box then you have this interface green color box you can see that the interface is Dolphin So basically what Dolphin is is called performance dynamic object oriented library for finite element computations it's a basically C++ Python library and it is a primary user interface then you again have other co-components like UFL FFC USC These are the core components These are the libraries which are part of the Phoenix application but then there are external libraries like NumPy Many of you are familiar with FETSC, Utilize, CLEPSE, UMFPAC and so on VTK and all this VTK is a particular value of the image So you have a lot of these libraries we have to import so as to get the limits Now what is the advantage of Phoenix Phoenix basically is an automated FEM Automated FEM means basically as I told you earlier that we have elements and we try to find out we have nodes and there we basically we find out the solution of those nodes so what happens that we make all these matrixes of form basically then we use Phoenix but however when you are using Phoenix don't have to coding then you don't have to make all these matrices and all that and using the weak form it automatically gets in that particular thing then we have this automated evaluation of original forms automated finite element assembly automated adaptive error control which has so many other advantages these are the type of elements when you discretize a geometry a 3D geometry a line geometry or a surface geometry these are the type of elements which are used in Phoenix then some more elements are here presented here as you can see how we will do one this thing we will do one case study we will talk about how the Phoenix is applied like how the Phoenix works basically we are solving a Poisson equation what Poisson equation is basically that it is a kind of a any kind of equation which is there and you are putting some heat like you have to take a plate and you heat it from the center how the heat is going to get distributed on to that particular plate how the heat temperature distribution is that can be basically some form of Poisson equation it distributes so that can be easily it is easily we can solve that Poisson equation and it can easily get results then you see like and then you have the weak form basically that is the second order equation to lower it down to a first order equation and you solve it this much amount of coding is there this is a Python code and this much amount of coding is required and you get the results basically how the temperature distribution is what happens then again this is the linear equation you take a beam you put some load in one end and you fix the beam on the other side how the beam is deflecting all the behavior is that can be basically solved using this linear elasticity coding and you can show it to you can get results for this also this kind of coding would be required it is again a pattern based and when you implement all these coding sort of programs then ultimately you get this solution like how the beam is deflecting you see how the beam is deflected when I fix it at one side and putting the loading on the other side this is the some of the work done by us like this is the fuel assembly it is a nuclear fuel basically we try to find out the natural vibration using this fuel assembly so what we do is this is the fuel assembly whole geometry you can see these are the ones which are tried in the triangular these are called fuel kits we try to find a natural frequency and we have done that using Phoenix this is the approach we have used for the algorithm that we have used and we have tried to find out this solution this is the fuel plates which are there we have tried out the natural frequency what the frequencies are for each particular plate for a given boundary condition if the plate is fixed on one side how the plate will be what will be the natural frequency all those things we have found out using this Phoenix software some more results are here can be seen so this is somebody that finite element analysis is one of the most widely used analysis techniques in engineering technology applied sciences and it is a popularity is expected to grow many fold in the coming years the method is a mathematical tool for solving partial differential equations literally referred as PDEs Phoenix is a powerful open source package for solving PDEs and was created with an aim of automated solution to mathematical models to finite element methods allowing easy reproduction of mathematical formulation and rapid implementation of high performance solvers here we have seen two kind of case studies where we have used Phoenix to find the solution one was kind of a heat equation and second was a last 50 equation so that's all these are the references I have taken literature references are the source codes for the images I have already made in the slides so thank you thanks a lot so could you also share your camera yeah yeah we can initially face some tech issues and great job solving it and getting back on the set so quite an interesting topic are you able to hear me yeah I am able to listen able to hear you so I think we have quite a few interesting questions you are out of time maybe you could answer them yeah I just had one question myself so what's the difference between sweeper and Phoenix which pipe SFP pipe there is a module sweeper that's also for FES actually basically first of all I didn't name the name of that particular software talking about Phoenix and sweep pipe sweep pipe SFP SFP pipe I am not aware of this particular software basically again it might be an operating source package basically there is not much difference basically what happens I will tell you what happens is that every software which is made in a university level it has an objective what kind of problem it wants to solve for example if you take any commercial answers you might have heard of it from that me and any other software like this thing console so generally what people see is console that particular software is very good for solving multiplicative problems like even answers can solve anything it has a powerful potential for that also but now this console which is a commercial software is widely used it's a very multiplicative tool it's like different fields interacting with each other like a fluid hitting of a structure or like a sound propagating through a solid or liquid media those kind of problems are a bit easier to solve in commercial multiplicatives compared to answers however it's a very difficult problem very difficult problem like I told you about number of elements you want to different times if you have like in millions like in 10 crore or 100 crore like that then a console is very difficult it has a potential I am sure but what we have seen in our offices like we generally go for answers for this kind of problems which are heavy problems like you want to model very intricate particular areas of volumes or lines so every software which is so there are a few to insert but I think we are on time we need to be really happy to go ahead and explore Phoenix so thanks a lot for the section it was great thank you a lot