hello welcome to basics of EFI a part 1 this is a 8 week course each course we will have six modules one module on each day Monday Tuesday Wednesday Thursday and Friday so overall we will have something like 48 lectures each lecture will be something like 20 to 25 minutes and and then at the end of the H each lecture we will also have an assignment which you will be expected to do most of these assignments will have multiple choice questions so please do those questions even though they are multiple choice to answer those questions you will have to have some good understanding of the theory of finite element analysis so it is important that you listen to these lectures carefully and work accordingly the book which I will be heavily relying on for this particular course is introduction to the finite element method and the author of this book is je and ready it's a very popular book and the strength of this book is that it's extremely easy to understand finite element analysis or finite element method is a technique to solve differential equations and it's mathematically very intensive so a lot of books in area of finite element method you will see that they are extremely heavy on mathematics and in that process sometimes the understanding part of the book through its language and through explanations is not that clear so the advantage of the book by Jane ready is that it is easy to understand and yet it does not compromise on the mathematics of the course so that is the book I will be referring to and then of course we will have examinations mid semester examinations and also and some exams and that is how we are going to do in this particular so this is part one of the course these eight weeks and then maybe in July onwards we will have a new set of MOOCs so then in that particular course I will do the second part of finite element analysis what you will learn in this first part will be basically the theory of finite element method as applied to one-dimensional problems and so you will understand how complex geometries are broken how they are discretized how our assembly element level equations written how our assembly is done a lot of stuff about terminology how do you apply boundary conditions what our initial value conditions what our boundary value conditions what is what are the different mathematical approaches for this and I think if you have a good understanding of 1d type of problems then going from there to 2d type of problems is relatively easy that is not a big jump so most of the conceptual you know understanding in context of finite element analysis it starts right away from 1d problems so we do not have to think that 2d problems are 1d problems are simple 2d problems are complex and 3d problems are super complex if you get a good understanding of 1d then going from 1 D to 2d the only additional step is how do you assemble elements in to do a two dimensional systems and then from two dimensions to three dimensions actually that transition is fairly straightforward because there is nothing new when you go to three dimensions so so that is so so even if you are planning to do the part two of the course which will be after June or July if the plan goes correctly then having a good grip on that particular course will very strongly depend on how we treat the subject at this level so that is the prelude so once again I welcome all of you to this particular course and I hope that you enjoy this course and learn something worthwhile and useful because finite element analysis is now very extensively used in almost all areas of science and technology and engineering so you can be a mechanical engineer or a civil engineer or an aerospace engineer you can be a material scientist you can be a physicist you can be a chemical engineer and virtually all areas of engineering in science in some form or other Fei is used because what Fe allows you to do is to solve complicated differential equations you know that suppose you have a function you can have any function and differentiating any function is a pretty straightforward process but integrating any function any our general function is not that straightforward because the way we have defined integration or integral of a function is we actually consider it as an inverse process of a derivative so suppose you have X square and I want to integrate it with respect to DX then the way I do it is that I know that derivative of DX derivative of x cube is three X square so then what I do is I know this information so I know that the derivative of cubic of X is three X square so I say that okay if that is the case then antiderivative of three X square is X cube and from here I infer that integral of this will be X cube over 3 okay so what this basic concept means is that if I have to integrate a function I have to basically I am trying to find its antiderivative which means that I have to figure out a function whose derivative is this thing and finding out that function is not a straightforward affair but if I want to differentiate any function that is pretty straightforward so the point what I am trying to make is that when we have differential equations which actually capture several physical phenomena for instance there is a heat transfer happening across a body then the way you capture the physics of that body is through a differential equation you have a body suppose this is a body and I am putting some force on it and I want to calculate stresses and strains in this body then again I have to write equations of equilibrium in differential form so again I have a partial set of partial differential equations which govern that phenomena and each time I have partial differential equations or ordinary differential equations I have to solve for unknown variables for instance if I am trying to bend this pen the unknowns may be displacements in the pen at different points in space and so those are the unknowns and the way I solve for those unknowns is I have to integrate the partial differential equation which governs the bending phenomena of a pen and as I explained in general integration is a much more difficult and complex you know non straightforward process than differentiation so finding integrals of she'll differential equations are ordinary differential equations in some cases it is easy and straightforward in most of the cases it is not easy and not straightforward so finding exact integrals of complex functions is not straightforward so then what we do is that we try to solve these equations not in an exact sense but in an approximate sense okay so we will say okay we are okay if my solution is not 100% accurate but maybe 99% accurate so we use some approximate methods to solve or to integrate these equations and that is their techniques like finite element method come into picture okay so I will give you some couple of comments on finite element method so so the first thing is that this method helps us numerically integrate and we will explain this term numerically as we walk through this course integrate or I can also call it solve differential equations now these equations could be PD's partial differential equations or they could be ordinary differential equation does not matter these equations could also be linear or non-linear it again does not matter a very big advantage of finite element analysis is that it is very general and because it is general it is also very flexible what do I mean by that so if you have a particular ordinary differential equation or a PD you can integrate it under some special conditions there are several ways right you have this variable separable kind method and so on and so forth so there are several methods but each method of integrating a partial differential equation is applicable to only a limited set of conditions okay it is only applicable to a limited set of conditions but when we start using techniques like finite element method this method is more or less insensitive to the type or form of the partial differential equation because the philosophy it uses and you will see it as we walk through in detail the philosophy on which it rests on is very generic so it does not matter whether the solution is of a equation is of this type or some other type whether it is homogeneous or non homogenous if we understand the overall theory of finite element method reasonably well then we should be able to solve or integrate these equations which differ with define or capture the essence of several physical processes fairly with a fair amount of accuracy and reasonless so this is very general and because it is general I can use the same method to solve a heat conduction problem I can use the same method to solve a thermal diffusion problem I can use the same method to solve a problem in computational fluid dynamics or do stress analysis or do dynamics or understand vibrations or understand the physics of the universe and so on and so forth so because its general it is also very flexible and another advantage is that it is highly systematic highly systematic and structured process what does that mean so a lot of times when we try to solve differential equations we make guesses right this gaseous solution and then we try that solution by plugging it back into the equation and see whether the equation is satisfied or not it is satisfied get satisfied or not this type of a process where you have to in guess or view you have to intuitively figure out the solution of a differential equation this type of a process cannot be easily implemented in computers because a computer cannot guess solutions okay in context of Fe a we do not rely on this intuition but the method is very systematic very structured and there is a very well-defined algorithm which drives the whole approach so if you have a very systematic and structured process then you can easily automate it it is easy to automate it and as a consequence you have a problem and you have a very generic code that generate code for stress analysis can solve how a pen bends or how if I am stretching this strap of this watch how does it deform or if I am bending this plate and it is deforming so the same code the same software is able to solve a very large number of problems and the variety of problems it can address is not limited by size or shape or material properties or things like that so it is easy to automate because it is highly systematic and it's a very structured versus and as a consequence of these things that it is very general and also because it is highly systematic and structured process it is used almost in all areas of science and technology so ok so that is there so now what we will talk maybe over next 5 7 minutes is give you a very basic overview again this will not make you experts in finite element method but it will give you a very basic overview of what is the overall philosophy of finite element method so the first step when you are trying to use finite element method so let us say that I have a let us say this is a beam and I am trying to bend it ok I am trying to bend it and I have to understand how this when I apply a force F on this beam and let us say the beam is rigidly fixed here and I am applying a force here this beam is trying to bend and when it is trying to bend I have to figure out that at each point along the beam what is the displacement in X direction in y direction and in Z direction whatever so and I want to let us say solve it using the finite element method so the first process the first step in this entire process is that we develop develop a set of governing equations this step is invariant in the sense that whether you are trying to solve the partial differential equation in an exact sense by directly integrating it or you are trying to use finite element method to solve the thing this step has to be there so you have to develop a set of governing equations for instance in this case of the beam what is it that we are trying to do we are applying a force and as a consequence of this force this structure is bending so what I am trying to develop is a set of equilibrium equations because there is mass here there is stiffness there are forces and are the laws of Newton tell us that this thing will bend and move only if external forces on each small element of this material is not zero so I take a small piece of material I see what all of what are all the forces on it and then I develop an equilibrium equation for this so that is one thing then using laws of geometry I try to figure out what is the relationship between the curvature of the beam and the displacements in the beam so those so first set of so first set and at least in context of this beam first set of equations would be equilibrium equations second set of equations would be strain displacement relations okay so equilibrium equations are basically statements of Newton's laws of motion strain displacement relations they are basically driven by geometry okay and the third set of equations will be how stresses and strains are related stress strain relationships now this P set of governing equations are applicable to let us say solid mechanics okay but if our problem is different say it involves heat transfer then these set of equations will differ and we have to develop a different set of equations which are applicable to the heat transfer process but regardless we have to we always start by developing a set of governing equations once we have done this then we what do we do we solve these governing equations you know so we solve these and before I go deeper into how we solve these equations using finite element method we will see what are we will have a very quick overview in terms of what are different types of methods to solve these equations so the first way to solve them is exact solutions okay so one way we can solve is we get an exact solution of these differential equations so what is the meaning of exact solution it means that my solution has to do two things it has to a satisfy governing equations exactly it has to satisfy an at all and at all points in area of interest okay so that is the first thing that whatever solution I have suppose I have I get a solution that displacement equals a X plus B X square plus C X cube plus D X 4 and so on and so forth this solution if I plug it back into my original differential equation then the left side of the equation and the right side of the equation they have to match exactly for all values of X which are which lie in my area of interest okay so that is the first condition that is the first condition the second condition for an exact solution is that all the boundary conditions are exactly satisfied okay so there could be a situation that the differential equation is exactly satisfied but the boundary conditions are not in that case the solution may not be exact there could be another situation where some boundary conditions are satisfied and some at some points the differential equation is satisfied but all boundary conditions are not satisfied or whatever so all the boundary conditions have to be satisfied and the second one is that the governing differential equations which we are interested in they have to be satisfied at all the points suppose I say that my displacement suppose I guess that the displacement in this vertical direction if I apply a force is you know C 1 X plus C 2 X square plus C 3 X cube plus C 4 X then at all the points in the beam at all the points in the beam that equation has to satisfy the governing differential equation which you know supervises or overseas the behavior of the beam at all the points so that is there now so if the solution is exact then our error will be 0 ok the other thing so that is the first set of solutions but then in most of the cases unless our problem is very simple these exact solutions are not possible so then we go for approximate solutions okay and there are broadly speaking two categories of approximate solutions first one is analytical solutions and then other one is numerical solutions now in approximates in in exact solution the error and what is the error the difference if you have a governing differential equation the difference between right-hand side and the left-hand side that difference will be always zero if the difference is not zero then that is regarded as the error okay that is the defined as the error of the solution in exact solution for all points the error is zero in approximate points the error is not zero at most of the points but what we say is that in this case one way what we try to do is that we will say okay if I have an error okay and I multiply it by some weighting function by some function at this point of time we will just call it some function some weighting function we assign it some weight and we integrate it and suppose the beam we are into if it is beam then we integrate it over the length of the beam then the weighted integral of the error is zero this is one strategy to achieve an approximate solution so point by point the error may not be 0 but error may be a little bit high at one point a little bit low at one point and if you add up all these errors after waiting you know then that is the integral right that is the integral so so this is this is Omega it represents o domain what domain means the area of interest the region of interest so if I integrate this error in a weighted sort of V and I will use this term weighted again and again and slowly it will become more and more clear to you what it means in a weighted sort of way if I integrate this error over the domain which means the a the region of interest then that error is going to be zero okay so then these approximate solutions there lie in two categories numerical and analytical and in analytical solutions we get in plain language we get some formulas as solutions ok we get some sort of formulas so maybe the deflection will be some sort of a series expansion or it may be polynomial expansion a series or a sinusoid you know this harmonic series or some sort or an exponential series so there will be some relationship between for some relationship some formulaic relationship for the unknown variables in which we are interested in in numerical solutions we do not get formulas but we get numerical values at different points in the region of interest that is the domain in domain and once again what by domain I mean the region of interest in which I am trying to find the solution okay so suppose I have this piece and I am applying some force here you know if I tried if I apply a force here and if I am trying to develop an analytical solution I will get some sort of I will have to develop some sort of function which will explain or which will define how UVW suppose displacements is what time interested in how v uvw are changing with respect to XYZ in this cone but if I am using a numerical solution then I will not have this kind of a formula but rather I will have maybe at this point stresses three point one at this point displacement two point seven so I will get numerical values so the finite element method is a numerical approach it is a numerical approach another approach which is numerical in nature so to approaches are popular one is Fe a other approaches boundary element method or Fe M it is also called analysis and the third one is finite difference method these are three methods okay one more thing so we can do approximate solutions even using this approach so we said that approximate solutions in approximate solutions there are two categories analytical and numerical and in both these approaches what we are trying to do is we are trying to have a weighted integral of the sorry weighted the we weighed the error and we do a weighted integral of the error we add that up over the domain and then we equated to zero so that is what we do we will learn what this weighted integral is later so you have to be patient but that is what we are doing another way to Phi to approximate solutions is to define so this is method 1 this is method 1 the other method method 2 is known as finite defy night difference method so actually I should have I should erase FDM from here and here we do not do the weighted we do not calculate the weighted integral of the overall system rather what we do is that we express derivatives as differences okay so suppose we have to calculate del u over del X then what we do is we calculate the value of U at point one at X is equal to x1 you calculate the value of U at another point where X is equal to x2 and then we say okay the partial derivative of U is u 2 minus u 1 divided by x2 minus x1 so once we do that we very rapidly convert these partial differential equations into algebraic equations and then we can solve those so that is another function so we have an integral approach this is method 1 we have another one method 2 finite difference method in method 1 we can have analytical solutions or numerical solutions method 2 is finite difference method and that is only a numerical method so this completes our first module and we will again meet tomorrow for the next lecture Thanks you