 Before going to various approaches in finite element methods, let us see how the finite element method works. So, as I mentioned, the initial step is, of course, now what we are talking is, we have already seen the mathematical modeling, so the problem definition is done, data collection is done, then we have already defined the domain and the conceptualization is done and the mathematical equation already goes. From there only the numerical method starts, so we know the domain. So, once we know the domain, say what we do, we can discretize the continuum with the elements interconnected at nodal points. So, you can see that here, we are having the domain like this and then this junction point is called node and this is called an element. So, you can see that each element is connected with respect to number of, if you put the numbers 1, 2, 3 like this, you can see that there will be, say accordingly, we can connect using the nodes. So, the junction between the elements that is called node and then we can connect this with respect to the elements and then we will get the domain. So, you can see that, say, if say for example, let us just consider rectangle domain like this. So, the numbering, say let us assume that we are using triangular elements. So, if we use the triangle elements, then we can see here what we do, we just say we can put this itself, this problem itself we will just consider here. So, what we will do, let us assume that this is triangle elements are there. So, what we do, we can just number like this 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. So, like that we number all these, the nodes, so these are so called nodes and these are called elements. So, element number 1, so this is 1, 2, 3, 4, 5, 6, 7, 8 like this. So, either a clockwise fashion or anti-clockwise fashion you can just number, so that why I am saying clockwise or anti-clockwise is that, so the computer as a logic, computer cannot just go randomly, so for that logic only we are putting either clockwise or anti-clockwise fashion. And then we say if one element is considered, say for example, element number 1, so we can identify, we means the computer can, we know we can easily identify, but computer will be identified by this nodal number, so 1, 6, 5. So, let us assume that this is say 4 meter by 4 meter domain, so this is 4 by 4. So, if this is, you can see that this nodal position, this is 0, 0, this is 1, 0, so this is 1, 1, so this is say 0, 1. So, if you give, so now element number 1 is identified by this nod number 1, 6, 5 and then we can write what is the position of the node, node number of 1 is 0 and this is 1, 0 and then this 5th one is 0, 1, so this one is 1, 1, so like this we can identify, so we means the computer, when you write a computer code, the computer is identifying, so the total domain is in terms of the elements like this and each element is identified with respect to this so called nodal points or nodes here, so 1, 6, 5, nodes are used for identifying the position and then we give the exact value, so the nodal position, so that now the computer after giving all this data, the computer can identify the domain and its details. So, this is actually a part of the preprocessing, so once the domain is known, we can just define, we can discretize it, so and then we can put the nodal numbers, element numbers and then we feed as a data set to the computer the element number and corresponding node number and then nodal values, so this is the way we discretize the domain and of course then you will ask how I know that say what kind of discretization where it is 5 elements, so you can see that the same domain we can use, we can discretize even to 2 elements or 4 elements or 8 elements or 16 elements or 48 or 96, so many things can be done, so you will ask how I know that how many elements, so this is coming through practice, so once you learn, so initially you may have to say for the given problem depending upon how much accuracy is needed, you can say identify, initially you may just use a coarse mesh which is so called coarse say with less number of elements and then you can run the model and see how the results are, then you can go for final mesh, but once you become a modeler then for the given problem you can easily identify yourself, how many elements are needed, how the discretization should be done, whether you can go for this kind of measure that kind of mesh, so that is the first step, so discretize the domain, the continuum elements interconnected at 167, yes this is 7, yeah this is 5 is there 167 only, yes thank you, so like that we can identify and then they say the first step is discretize the domain, elements interconnected at nodal point, so then the second step is once it is done discretize the domain is discretized, then the next step is selection of interpolation function by representing the variation of the field variable over the element, so you can see that here say the total domain is mathematically represented in terms of the governing equations, this flow equation, transport equation, Laplace equation, Navier-Stokes equation any kind of equation, then we are discretizing the elements like this, so what we are representing is that we are say we are putting certain assumption that say it is so called the function called interpolation function that is representing the variation of field variable over the element, so we can have different types of interpolation functions, it can be sine function, cosine function or it can be the triangular element that you know that a minus x plus b minus y like that different types of function we can define, so that is so called interpolation function, so we use this interpolation function first approximate how it is done, we will be explaining later, so that way we represent so the representation of the field variable over the elements and then we find the element properties, element properties means if it is a triangular element how the variation is taking place, how the degree of freedom that means how the variation is taking place within that element, so third step is find the element properties and then fourth step is assemble the element properties to obtain the system of equations, as I mentioned so each element we are applying the governing equations and then we are approximating with respect to the elements and then we are writing the equation for the element and next step is we assemble each element so that with respect to the connectivity the computer can identify the element and then it can assemble, so then next step is once this is done, next step is we know that the problem is behaving according to the boundary conditions, so like if it is say ground water flow problem, so if you know the head here, let us say here 10 meter, here 5 meter then these are the boundary conditions, so according to the boundary conditions only the problem is the problem we are solving, so the next step is application of the boundary condition and then once application of the boundary condition is done then we are having most of time we are having a system of linear equations, so we just solve that system of linear equations and then we get the variables it can be head variation, it can be velocity variation or it can be concentration variation or whatever type of problem we are solving and then if any additional calculations to be done then you can also write code for that, so for example in ground water flow once you calculate the head variation then next step is you want to say to solve for transport equation you want to find out the velocity variation so that you can just apply the Darcy's equation v is equal to k into i, i is the head variation with respect to distance, so that we can additional computations can be done, so you can see that, so in finite element method different approaches are there, so in all the approaches this kind of a systematic approach starting from the domain and then solution and additional calculations of this way we are having a systematic approach in the finite element modeling, so here I have shown here a flowchart, FEM procedure it is taken from Baate KJ to finite element procedure is the test book by Baate published by Prendice Hall of India, so here we can see that as I mentioned this is already discussed in the morning lecture also, so here we are having the physical problem and then the mathematical model, so the governing equations then geometry and all the material properties etc etc and then we can we are looking for the finite element solutions that means initially an element finite elements then mesh density we will be looking at a solution parameters and then representation that means the additional loading or the excitation and then boundary condition etc and then we assemble the element matrix then we solve the element matrices and then of course many say to make a very good code very good model we may have to come back and then also we may have to do the solution or the process with few more iterations so that we will get a better model, so this shows a typical flowchart for the finite element modeling, so now as I mentioned earlier in finite element method there are so many approaches are available, so this depending upon the problem which you are solving or depending upon the your knowledge or depending upon the area of interest the particular approach can be selected for the given problem actually as for the finite element method is considered the origin is say based upon the the applications in civil engineering you know all of you know all of your civil engineers you know that what is a truss, so in a truss you can see that what we are having say here if you if you analyze a truss then most of the time what we will be doing say here there is if this is a truss then you can see that there will be a number of elements depending upon any I am just drawing like this, so here we will be having number of elements and then here there will be supports like this and then there will be number one two three like that you can number, so this is a truss or if you consider a pipe network all of you know what is a pipe network here say we are having a pipe network like this, so in all these problems or even in electrical engineering we can have an electrical resistance network like this, so you know basic ohm low ohm slow and other things, so non electrical network in all this network you can see that here we are having in for the truss here say here the joints there number of joints are there then we have a number of members like this and then we connect it together and then either we rivet or we will using a tan bolt we tie it and then it becomes actually in the beginning it is number of small small members finally you can see that once you put it everything together it is a single member, so that is the same principle is what we are using in finite element they are having small small elements but when we connect it together it is a single domain so that is the basic principle, so actually you know that this truss and this type of structures we are using for many many centuries, so actually the beginning of say the finite element method is based upon this kinds of truss or pipe network since the people thought say 1950s especially processing weeks and his team and all others, so then there are so many big names in finite element method, so they thought that this here when you are analyzing a structure you know what is the method we use for general simple analysis method of joints or method of sections, so if you say for example if you can see method of joints you can see that each joint we are taking and then what is sigma h sigma y sigma v and then we are analyzing, so similar way then they thought why we cannot say based upon this principle why we cannot formulate a method, so then they formatted say initially for this kinds of problem problems like truss problems like pipe network, so here in all this you see it is a truss or a pipe network or an electrical network like this there is a physical principle behind this you can see that pipe network is concerned mass balance how much is coming here how much will be taken and how much will be the flowing out, so with respect and then a node is concerned what is going in what is what should go out, so that is the physical principle of this and the truss is also considered the forces sigma h is equal to zero sigma v is equal to zero when we analyze, so that is the physics of the problem or the physics behind a truss, so when we are analyzing we like this then people thought of course this is the physics of the problem, so that principle we can utilize to analyze the system, similarly an electrical network like this you can see that we can use ohms law, so ohms law you can see that say with respect to voltage and electric current then the resistance there is the relationship is there so called ohms law, so we can utilize directly that ohms law, so that we will be we can we get a we get a physical principle connecting all the all the resistors like this all the members in a truss or all the pipes in a pipe network, so that is the physics of the problem, so now what we can do like in finite element as we have as I have mentioned we can take each member see if we are analyzing a pipe network like this then you can consider the junction like this and how much is flowing in how much is flowing out that you can find out or if you consider truss like this we can use the principle like sigma h is equal to zero sigma v equal to zero, so the first approach actually the finite element method has been derived based upon this essence of this kinds of problems, so civil engineers can we can boast that the original finite element method is from civil engineering, so here the direct approach this is the first approach which has been designed in 1950s the end of 1950s and at the beginning of 1960s, so the direct approach says like network analysis pipe network electrical network or structural framework analysis we derive the element properties from the fundamental physics as I mentioned either ohms low or the mass balance or the force balance and then say and depending upon the nature of the problem we can set an equation, so then we can easily so like here using that principle we can analyze each joint or each which is in finite element method we call node, so not a point wise we can analyze, so here it is why it is called direct approach since it is very simple we have direct principle is utilized in the analysis and that is the working principle behind the finite element method, so the direct physical reasoning to establish the element equations in terms of the pertinent variables, pertinent variables can be flow in a pipe network or the forces or the stress in a truss or the current flowing through the network like that, so the direct physical reasoning is there, so that is we can directly utilize, so the essence of finite element is coming from these kinds of problems, so this is but there is you can see that there are certain limitations say for example if you want to solve a ground water flow problem it is very difficult to how this can see when the some equations can be utilized but we cannot separate it like this, so we cannot just utilize this direct approach for the solution of all the problems, so we have to then the mathematicians have derived a number of theories for so many centuries, so these some of these theories came into the help of the those who developed the finite element method, so one of the second the important method one of the most important method used is called variational approach, so here the variational approach they say instead of the we are solving the original governing equation we derive an equation so called variational using a variational calculus we derive a special equation called variational equation that is called variational formulation, so here we find the unknown function and then we extremize or make stationary a functional that means based upon that if the governing equation say Laplace equation any kind of equation we can derive a special function called variational function and then instead of solving that original equation we will be solving this we will be maximizing or minimizing or we make it stationary with respect to that special function so that kind of approach is called variational approach, here the advantages are more complex problems and boundary condition can be accounted lower order derivatives that next slide I will show how it becomes lower order derivatives than differential operator we can derive and then this relies upon the calculus of variation, so here I have also the procedure is we discretize the domain we substitute the trial functions into the functional that means the the variational function as I mentioned and then we differentiate the functional and then equate to 0 so that it will be minimized, so the system assembly that means minimization means minimization of the error so that is what we are doing, so then we assemble the system and system equations are formed and then apply the boundary conditions and then we can solve for the unknowns, so variational approach is very much used especially structural mechanics, structural engineering most of the time this variational approach is since you know that most of the time we will be solving fourth order equations, so that fourth order equation can be reduced to third order or third order equation can be reduced second order by using a variational functional and that functionally is used in in the in the finite term analysis, so this variational approach is very much used in structural mechanics especially in civil engineering, so here you can see that the variational approach say for example if you consider the Laplace equation you all of you know what is a Laplace equation so del square 5 is equal to 0 in a domain omega, so there will be of course any mathematical model will not be complete if you do not specify the boundary condition say here the boundary condition can be pi is equal to non-value some locations and then its normal derivative del phi by del n is equal to some value non-values so this can be the boundary condition, so here for the second order equation we can derive a functional called variational functional, so here this can be mathematical you can see all this in advanced mathematical textbooks given like crazy book number of books are available, so how it is derived I am not going to since within one and half or I am more interested in applications and then for more discussion about the methodology, so how we are getting you can look into advanced textbook like here textbook by crazy, so here the variational formulation here this corresponding actually this equation here is correspondence to this Laplace equation and then instead of solving this equation the Laplace equation we can solve this equation, so that is what is so called variational formulations and then the procedure is essentially same instead of solving the governing equation we are solving the variational equation, so that is what we are doing in the variational approach, so this is very much used in civil engineering especially by structural engineers, so now the third approach is called energy balance approach, so here what we will do we translate with say a relation that holds over a finite region, so it can be generally it can be energy balance or heat balance or the kinetic energy or from the first law of thermodynamics that means global energy balance, so here actually generally this kind of approach is used by mechanical engineers not related to structures but where heat and thermal problems are there, so there they utilize this method this also but is not very much used in our types of water resource problem, so here they use the energy balance is that is the essence of the model, so as I mentioned say some physics of the problem is represented by certain relationship, so that is obtained through the global energy balance, so that is the essence of this methods so called energy balance approaching finite element method, so now the next method which is we very much use in finite finite element method in a water resource and environment is so called a waiter residual approach, so here the method of waiter approach is a technique of obtaining approximate solution to linear non-linear partial differential equation, the advantage is that there is variational approach we have seen we have to derive a special equation which will represent the governing equation mathematically but here we do not need these kinds of equations we can directly attack the problem, so here the procedure is listed here we first discretize the domain considered with the suitable elements then we assume the general functional behavior of the dependent variable that means in head variation in a kiffer or concentration variation for transport problem then in some ways to approximately satisfy the given differential equation over the elements, so how it is done I will show in the next slide, so substitution of this approximation into the original differential equation and boundary condition results in some error called a residual, so as I mentioned we are discretizing the domain in terms of elements, so we take each element and then we are applying that the governing equation to that element, so and then say we know that actually the governing equation is given in such a way that it should really represent that the problem within that element but we here since we do know the actual variation of the variable we assume certain in terms of certain functions like as I mentioned it can be polynomial, it can be a sine function, it can be cosine function, so within that variable say it is just like a plus bx, so it can vary in linear variation or a plus bx plus cx square, so you can see that it is a quadratic variation, so a plus b plus c a plus bx plus cx square plus dx cube, so it can be called a cubic variation, so like that how so for more accuracy we may need for higher order elements, so we how within that element we can represent generally in terms of a polynomial or a function like a cosine function or sine function and then that function is used to represent the variation of the particular variable within that element and then using that function in the governing equation we will approximate the governing equation so that the element elemental equation is formed, so substitution of this approximation to the original differential equation and boundary conditions, since it is an assumption that this is representing that variation so there will be an error, so what we do that error is so called error pseudo in the finite error method that error is so called the pseudo, so we try to minimize that error by using certain mathematical techniques so that that error will be vanished within that domain, so that is the essence of this method of vector residue, so the as I mentioned the finite error method is concerned the procedure is essentially same, so one or the here only so as we have seen the variation approach there is a variation function should be there but method of vector residue here we assume that it is represented in terms of certain function called interpolation functions and then we are representing in terms so that and we substitute that to the governing equation, so then the resulting system of equations are solved to get the approximate solution what we are looking for so that is the essence of this method of vector residue, so this approach is advantageous because it thereby becomes possible to extend finite error methods to any problem where there is no functional available, so we have already seen the case of variation approach we need a special functional based upon that only the method is proceeding but here it is not needed we do not need any special function we can just directly utilize we just put the a variation with respect to certain polynomials or any other function and then that we substitute to the governing equation and then we are approximating or we are forcing the error to minimum, so let us assume that we want to find out an approximation function representation of field variable phi governed by this differential equation l phi minus f is equal to 0 as in equation number 3.1, so the method of vector residue is applied in two steps as follows the first the unknown exact function, so here you can see that the variable is phi, so this phi is approximated by phi bar phi under bar here you can see that this is using the instead of phi we use a trial function phi bar where either the functional behavior phi bar is completely specified in terms of unknown parameters or the functional dependence on all but one of the independent variable is specified where while the functional which depends on the remaining independent variable is left unspecified, so we represent this l phi in terms of l phi bar, so that we can write this variation of the function that this phi is the varying function, so now instead of phi we are representing phi bar, so phi bar is an approximation for phi and this phi bar is again represented in terms of called interpolation functions, so that is as I mentioned there we utilize a polynomial, so a polynomial is written say sigma i is equal to 1 to m ni ci where ni are the assignment functions and ci are either the unknown parameters of unknown functions or one of the independent variables, so here and m functions of ni are usually chosen to satisfy the problem boundary condition, so here you can see that this ni can be a polynomial or any kind of function and then we represent this phi bar which is again a trial for the an approximation of the original phi and we write just like in equation number 3.2, so once it is done now this instead of phi we write now phi bar, so this is substituted back into the equation number 3.1, so that we can instead of l phi now the governing equation is l phi bar minus f but now we cannot equate to 0 since it is now it is an approximation, so there will be some error, so that we have to put it in terms of residual or so called r, so here the error is represented in terms of r, so l phi bar minus f is equal to r as in equation number 3.4, so where r is a residual or error that results from the approximation of phi by phi bar, so the method of weight residual seeks, so the mathematical procedure is it is seeking we want to determine the m unknowns we have seen in terms of how it is supposed to phi bar, m unknowns of ci in such a way that the error r over the entire solution domain is small, so all the details are written in the lecture note, so it is accomplished by forming a weighted average of the error and specifying that this weighted average is vanishing over solution domain hence we choose m linearly dependent weight varying function w i in such a way that here we can write we now for that element the domain we integrate within that domain l phi bar minus f w i d d, so this w i is so called an independent weighting function for the to reduce the error and now this we force to 0, so this now we are integrating within that first element then total domain, so within that element we force this to this error what will occur due to this we are forcing that to 0, so once r is approximately 0 in some sense then we can we can solve the system of equations and then we will get the solution, so the form of the error distribution principle expressed in equation 3.5 depends on what kind of function you are using, so once we specify the weighting functions it represents a set of m equation, so I will show you some of the governing equations how to approximate this then you will get more idea, so here a set of equation either algebra equation or ordinary differential equation to be solved for ci, so the second step that is the first step we substitute it and force to 0, so then the second step is then we solve equation 3.54 the unknown coefficient ci and hence approximate representation of the unknown field variable phi is obtained through this equation 3.2, so here this 3.2 is here, so we get that and then there are a variety of weighting error techniques because of the both choice of weighting functions or error distribution principle as I mentioned a number of functions we can utilize and then accordingly the how we utilize the function, how you define the function a number of sub methods are therefore method of weighting. So, one of the most commonly used method is so called Galerkin method in Fientland method, so the error distribution principle most often used to derive Fientland equation is known as Galerkin approach, so in the Galerkin approach according to Galerkin approach the weighting functions are chosen to be the same as the approximating functions used to represent phi. So, here we have seen that we represent that initially with respect to n i that means the previous equation you can see here we represent in terms of n i, so in the Galerkin approach, so what we do here we represent this n i directly as w i where we started to page, so w i is equal to n i for i is equal to 1 to m for the number of nodes or number of nodes which we consider, so we use the same function then it is so called Galerkin finite element methods, so this is one of the most commonly used method especially for water resource and environment related problem. So, initially we can form a local approximation analogous to 3.2 and Galerkin procedure for an individual element and then as I mentioned this equation is initially written for the individual elements and then we assemble the system of end up for the entire domain and then the functions n i are recognized as the interpolation function, so this n i which we are using it is called the interpolation functions and define over the elements and c i are the undetermined parameters. As you can see if you consider triangle elements then we will be defining in terms of say n 1 x 1 plus n 2 x 2 n 3 x n 1 x 1 plus n 2 y 1 or n 1 x plus n 2 y, so that this n 1 n 2 that c you know c 1 x 1 plus c 2 y, so c 1 c 2 can be formed after defining the shape of the element or the nodes are known this c 1 c 2 can be easily found. So, this is the essence of the Galerkin Feynman method. So, once we write now the system of equation can be written throughout the domain, so this is actually for the element for the element and then we integrate for throughout the elements that means now we have already seen we write 1 2 n number of nodes or n number of elements, so with respect to that nodal points we are having number of elements. So, if you just integrate all for all the domain so that we can integrate throughout the elements, so we can get a system of equation very similar to what is given for the element we get the equation an assembled equation. So, we have set of equation for each element for the whole assemblage then finally we can assemble for all the elements and the choice of approximating function n i should guarantee that inter element continuity necessary for the assembly process. So, you can see that if you are choosing a particular polynomial or particular function the one of the important aspect is that that the inter element continuity should be there that means when we are going from one element to another there should be continuity should be met since you can see that finally what we are doing is we are knitting together all the elements in the assembling process. So, inter element continuity is very important in this process. So, now by applying the integration by parts to integral expression equation 3.7 this equation we can do a integration by parts we can obtain expressions containing lower order derivatives. So, you can see that especially say for example when we deal with the ground water flow problems or any kind of that kind of transport problem you can see that some of the boundary conditions will be in terms of the flow variations and some of the boundary condition may be in terms of head variation. So, here you can see that say for example a simple domain like this. So, let us assume that this is a ground water flow domain and then the flow is taking place in this direction. So, here we know that this is 10 meter here 5 meter and let us assume here the rocky rock is on both sides or clay material where flow cannot go like this. So, flow is not allowed only flow in this direction. So, you can see that the condition will be here this when we discretize a domain like this. So, you can discretize the domain and then we can when we represent you can see that we can just say here the boundary conditions will be here the this nodes. So, this node is the head is known the wherever the head values are known this known this is known this is known and here you can see that this is so called a no flow condition. So, there is flux that means flux cannot pass through this points. So, this is also no flow condition. So, with respect to the no flow conditions you can this condition we can easily get through by just doing integration by parts for the for the given expression. So, you can see that say the if the flow equation the governing the if it is the ground water flow equations or the transport equation that kind of no flux condition or even the flow conditions can be obtained by doing an integration by parts. And then finally hence we can use approximating function with lower order in the element continuity. So, like this and finally the system is assembled for all the elements boundary conditions are applied and solve the system equation for the unknowns. So, this is the essence of finite unknown methods you can see various methodologies are there various techniques are there but for fundamentally used methods we have discussed here and this method of weight loss release as I mentioned for water and environmental problems method of weight loss review is one of the most commonly used methodologies. So, now let us see one two one one one one one dimensional problem and two dimensional problem corresponding finite element formulations and how we can solve. So, here we consider as an applications one dimensional in porous media. So, the transport the transport differential equation in saturated homogenous porous media we can represent by this equation r del c by del t by minus del by del x of this d is the expression coefficient c is the concentration ui is the velocity n is the porosity lambda is the radioactive coefficient r is the retardation factor and q is any source sourcing and corresponding concentration. So, this is the equation. So, as we have seen the one dimensional expression equation we can what usually the variable here is concentration. So, we represent c in terms of the interpolation function n i where n i is the interpolation function. So, n i is sigma n i x c i t and the trial solution we orthogonalize with respect to the residual over the domain and we integrate by multiplying this n i and with respect to this l c dx like this. Now, n n orthogonal condition to be satisfied so that this we can integrate and then we put it like this and we equate to 0 and finally, we can for the. So, if you do for the complete equation you will get a final equation a system of equation like this. Then of course, what we are discussing is so far on space as I as we have already discussed in the morning there is the finite difference is concerned the spatial variation and temporal variation. So, very similar a finite and a method is also considered we should we have to see the spatial variation the spatial variation is what we do through this type of discretization that is spatial variation. So, similar way we have to see how therefore, transient problem or unsteady problems we have to see how the variation is with respect to time is with respect to time is taking place. So, generally very similar way we approximate using some interpolation functions for the spatial variation the time variation also can be done like that very similar way. So, the time variation also can be done through an integral approach as we have seen for the spatial variation, but this is slightly complicated since one more already one integration over the space then one more integration over the time domain is done it will be complicated. So, generally what we do the time variation is represented with respect to a different scheme for a final different scheme. So, you can see that say this del c by del t. So, the previous equation we have seen this del c by del t term is there. So, we can represent this c t plus delta t minus c t divided by delta t. So, we will be representing these terms using a final difference term. So, only that time depending term. So, that will be representing like this either using a backward difference forward difference or a center difference depending upon the methodology which utilize. So, that will be generally used it and then we can just put it back to the equation and then methodology will be much simpler. We can of course, go for the entire time and space the finite element method, but for space also time also it will be slightly more complicated. So, generally we adopt a different scheme like this and then we can easily solve it. So, if you do that kind of approach here this del c by del t term here. So, actually here there will be a matrix p there will be a matrix k and then right hand side where that flux or any the non terms are there that we put the right hand side. So, we can write the equation like this. So, only here this is a finite element equation now del c by del t will be represented by c t plus c so, supersede t plus delta t minus c t divided by delta t will be represented and then we write the corresponding the matrix equations like this. So, what as we have discussed with the method of weight result or Galerkin finite element approach is applied and then we will be getting the final matrix like this. So, you can see how this del c by del t is dealt. So, here we see we can simplify like this in this form and then the solution after incorporating the initial and boundary condition is the unknown values of the concentration c at the nodal point at any time instance and then the values of at earlier time instance So, with respect to that means c t is known c at time t is known then we can find out the concentration at t plus delta t. So, that is what is done. So, here the initial conditions boundary conditions we can prescribe and then we can solve once this boundary conditions are substituted back to the form of the equation then we get a linear system of equations. So, we can solve it by either a direct method like a loss animation scheme or we can go for iterative schemes like loss illiterative schemes. So, I hope you know how these how to solve a matrix. So, these are given in advanced mathematical test books you can easily find out. So, now here I have just solved one simple example problem here and then of course, as I mentioned in the morning whenever we are developing a code we have to verify with respect to available analytical solutions. So, here for a concentration or contaminant transport problem there is a analytical solution given by Marino in 1974 published in general hydraulics engineering general. So, that analytical solution is available. So, here we considered 600 meter long actually this is a one dimensional problem. So, in one dimension 600 meter long we used as the domain. So, you can see that the various discretization. So, as I mentioned if you are at the beginning stage then you may use one course mesh like this then a final mesh or more final mesh. So, initially we used 25 meter length then 12.5 meter length then 6.25 meter length. So, as I mentioned you are having very final mesh then the accuracy will be much better. So, here you can see how the variation is taking place with respect to distance. So, the concentration variation. So, here you can see that say 60 days, 120 days, 180 days and with respect to various days it is plotted and then one of the line is for analytical solution. So, how it is behaving with respect to time it is shown here. So, delta T also not only the space variation the time step also very important. So, you can also use a sensitivity study with respect to time step delta T 1 day 2 days or 5 days like that. So, here delta T 1 day is giving much better solution. So, this is the way we can we verify with respect to available solution. So, as I mentioned the morning generally the which since we are using the final different scheme with respect to time there is certain criteria which you can utilize to have a better solution. So, one is so called core and number where core and number is defined as the velocity multiplied by the time step divided by the space discretization. Generally, as a mathematical criteria this should be less than 1 for stable solution better solution and then for the Peclet number is they defined as u delta x by dx where d stands for the dispersion coefficient. So, that should be less than or equal to 2. So, this gives better solution for this kinds of problems. So, this is one dimensional problem is simple one dimensional problem we have seen and then here another problem is two dimensional advection dispersion in porous media. So, you can see that the governing equation is given like this del by r to del c by del t is equal to del by del x of this equation this equation is given in the lecture notes also. So, here v x v y are the velocities and then if there is any source or sink or the with respect to that how the concentration changes and any is the porosity then this v x v y velocity variations. So, all these parameters are defined here. So, to solve this type of equation how we can utilize the in two dimension how we can utilize the binomial method. So, the as we have seen the previous case here the first stage is. So, we can represent the original concentration variation with respect to some interpretation functions like this. So, here we use the triangle elements. So, the triangle elements are considered we can write like this n i x y c i t plus n j. So, you can see that for a triangle element. So, this is an element. So, for element there will be three nodes. So, with respect to that three nodes the interpolation functions are described here. So, we can see the sample interpolation functions n i e n i n j and n k n i n j n k. So, we can represent with respect to area a is the area of the triangle element x is x y variation and a i b i c i other unknown coefficient one is the triangle domain is defined we can easily find out this a b c etcetera by using these equations here. So, now, we use this interpolation functions as we have seen earlier. So, these are used to represent the concentration variation and then we apply the Galerkin method. So, here the Galerkin equation we approximate. So, you can see that the Galerkin equation is written here in terms I know I will skip to the intermediate step. So, intermediate steps the very similar way what we have seen for one-dimensional problem using method of weight resolution we can write the the governing equations. So, first we write for the elements and then we the assemble for the total element number of elements and then we will be having the final system of equation like this. So, now, again a final different scheme is used for the time variation and then we can use say also here as I mentioned in the morning there can be implicit method explicit method for the time is considered since we are using here for a final different scheme for the time variation. So, very similar way they of course, we can we have seen that for the transport equation is considered we have to solve the we have to get the velocity variation at each element level. So, then only we can go for the transport solution. So, to solve the to get the velocity variation we will be solving a flow equation like this say for example, if it is confined aquifer problem then the governing equation is like this very similar way we can approximate this H which is the variable here H can be approximated and then we can get a finite element formulation for the flow equation is considered. So, this is for the flow. So, after solving this equation what we will be getting is we can get the head variation from the head variation using Darcy's law we can find out the velocity. So, that velocity values we will be putting back to the transport equation and then the transport equation will be solved. So, that is the procedure for solving flow and transport say especially in the ground water flow or the transport problem in aquifers. So, here as a sample problem I have shown here a rectangle domain where this is a confined aquifer system here you can see there is the head is 75 meter here the head is 70 meter and here we are having a pond with a polluted water and then that is polluting the aquifer system and the area is about 1800 meter length and 1000 meter width and the aquifer thickness is 25 meter and S storage coefficient is 0.0004 and then the say a simple grid with 60 knots and 90 elements are shown here and this area is a recharging area and here we are having a no flow boundary here we are having a no flow boundary and here we are having three pumping wells these are the pumping wells and these three are the observation wells and then we are having this well as now this black dots are observation pumping wells and this dots are the observation well and we are having a recharge well like this. So, this is a typical problem. So, you can see that as I mentioned the ground water flow it is very difficult to identify all the old stuff just like in the case of surface water. So, this is a typical problem which we want to demonstrate how it will be working. So, the typical finite element discretization using triangular elements are shown here and all the features of typical field problems are also and described here. So, the aquifer is recharged as I mentioned within that area and then lock meter dispersity we assume as 150 meter transverse dispersity 15 meter and rate of seepage from the point 0.005 meter per day and then the other parameters like delta x the coefficient that means d x and d y y are obtained like this and then the boundary conditions are 0 concentration here here the only the aquifer is polluted from the what we are taking place from these points and then this recharge well otherwise here it is 0 concentration on this upstream side and downstream side is kept it opened. So, that we do not prescribe any boundary condition here and here the gradient of the concentration is equal to 0. So, for this aquifer region we consider 3 pumping wells pumping 500, 600 and 250 meter per day and the recharge rate is 0.005 meter per day and the concentration coming through the aquifer system is through the polluted points which is having a 4000 ppm total 3DS and is also from the recharge well 1000 ppm that means how much is recharging accordingly the aquifer is polluted. So, what we do we have already seen the governing equations and then we have also already seen the the numerical modeling or the finite time modeling. So, we apply the boundary condition all the boundary conditions first we have to of course, solve the flow equation and using the boundary conditions and then we get the head variations at each nodal points and using that head variation using Darcy's law we find out the velocity variation and using that velocity variation we solve the transport equation. So, that we can find out the concentration distribution. So, then we can solve this for this problem is a time dependent problem. So, we can find out how the time variation is taking place and then we can solve for the flow and transport and then say for example, this prediction is after 10,000 days how the aquifer will be polluted. So, you can see that here the 1500. So, the initial concentration was 4000 ppm. So, that is you can see that it is distributed throughout the aquifer within 10,000 days. So, this is a typical problem which I have selected for this demonstration. So, now if you want to solve a real field problem how we can do a finite time modeling actually this is the problem which I am mentioning here this has been we have done as a project in while I was in Germany. So, here the problem here is say this is a city area actually in Germany actually about 80 percent of the water supply is obtained through ground water. So, most of the areas are sandy type of aquifers that means subsurface and then most of small cities they will be having their own pumping wells near to their areas and then they will be pumping out of that. So, the problem came to us was so the area is so called biblies. So, here the pumping wells are located in this area and actually earlier tanks in 1970 onwards there was a chemical plant here which was storing the trichloroethylene in some kinds of chemicals used for industry purpose that was storing in underground tanks here. So, what has happened is that without the knowledge of the company this tank has started leaking and then a big plume was formed and then it is according to hydraulic gradient this was just slowly moving to the in the direction of flow. So, actually the city has got a number of pumping wells which is supplying water to the aquifer system for the for the city from the aquifer system water was supplied. So, here you can see some small small lakes all these features we have used for this aquifer the for modeling. So, you can see that as such here it is very difficult since this area is extending like anything the aquifer is a large aquifer. So, it is very difficult to identify an aquifer system with boundaries. So, what we do we have some we have got some data from the the municipal authorities how the head variation is there. So, and then the head variation conduce they have given. So, and moreover in countries like Germany there is not like in India where only three months months soon and then eight or nine months of drought situation. But here most of the time say there will be rainfall say we intermittent in all the months and then the head variation will not be so drastic. So, this we checked for a few years how the head variation is taking place and then what we did with respect to two conduces here the these values we found that within the annual it is there is not much variation. So, we identified a conduce here we identified a conduce here. So, that values is we fixed it as such and then these conduces are known we then we identified we just put us a streamline like this and then finally we identified the the total the aquifer area which we want. So, here the problem was that what to the the problem came to us was here there is a condom and plume is identified they want to know how many days it would take to reach these pumping wells and then what kind of remediation process can be done for the given for the problem. So, that was the question came to us. So, we we had say I was working in University of Karlsruhe Institute of Dome Mechanics we have got our own packages of final model are there. So, but we used a package called ground water modeling system for the GMS called for the the the discretization. So, now the problem domain is defined and then the plume position is also known. So, we have to solve for flow then we have to find out the velocity variation and then we have to find out how the condom and transport is moving condom and plume is moving and then we have to identify certain pumping wells with respect to that we can if we want to go for pump and treat there is this is one of the commonly used method for remediation of this kinds of problem we pump out the condom and plume the condom and water from the aquifer system and then we treat it and then we disturb to some streams nearby. So, this is one of the commonly used technique. So, how many how much is to be pumped out for to remove this plume within certain 5 years or 6 years or that kind of time period. So, this this was the problem. So, then we analyze the aquifer system. So, actually you can see this shows the hydrogeology of the area there we a number of small layers are there, but we for the time being we approximated this aquifer system is to a two layer aquifer system like this and then the details are given here. So, two layer what is the hydraulic conductivity or the transmissivity variations that is already with respect to the pumping desk available we obtain the data. So, here there is C means in German it is lake this is C SWC means lakes. So, there are so small lakes are also that is interacting with respect to the aquifer system and then here this is the contaminant plume and here is the location of the pumping wells. So, and here there is a separate separation between the layers. So, these are the boundary conditions which we utilize for this problem. So, here 86.5 on the upper layer and bottom layer 86.25. So, like that the boundary conditions were prescribed and then you can see using the software called GMS we made it very fine mesh system actually this is about 20,000 elements we are used for the discretization. You can see that here I did not we did not use a uniform mesh, but we used a very say some areas here you can see our as I mentioned we want to identify how this contaminant plume is spreading that is our major intention. So, we used wherever that plume is there that area we used a very very fine mesh and wherever that not of our much interest then we used a coarse mesh. So, a coarse mesh and fine mesh and here are the actually our pumping wells are located that is by again here finer mesh. So, this is the finer element mesh which we used for this system and then by using this discretization and then using the boundary condition as I mentioned we develop the we use construct the finer element model and then we constructed the how the head to variation is taking place initially as I mentioned the head to variation we have to calculate. So, the initially this value is here the values are known here the values are known, but we want to find out total variation with respect to the various the conditions here with respect to various elements. So, the we found the variations the the conduce and then next step was we use the Darcy's equation to see the fine the velocity vectors and then use that velocity vectors in the transport equation and anyway that transport simulation I have not presented here. So, that transport simulations are used to find out how the system how much the spreading takes place how many years it will it will reach the the pumping wells from the present location and then if you go for any kind of the the kind of say we want to go for any kind of remediation how many pumping wells we can utilize at particular locations. So, that this much is a pumping rate for this many years. So, finally what we can get is we can find out how many years are required to the remove the entire plume or at least stop the plume so that it is not further moving to the to the particular wells which we which is pumping the water for the municipal corporation. So, this is this is the way which we construct finite element modeling and now for this particular session. So, here we have discussed the basics of the finite element methods. So, its application to various water source problem and environmental problems we have discussed. So, you can see that one of the most commonly used method is so called Galerkin finite element approach where we are utilizing method of heterosodial which is one of the symbol most methods. So, that is say if as I mentioned before if you are going for variational approach we should you have to find out the variational function and then it is slightly more complicated and here you can see that this method compared to the finite difference method this is much flexible. So, and more accurate and then method is say very simple means I will not say very simple mathematically, but say the to understand it and to work with it is much better and we have discussed with respect to the triangle elements, but we can have variety of elements depending upon what kind of model whether we are going for three dimensional model, two dimensional model or one dimensional model we can have a variety of elements. So, in one dimensional model we can have line elements and then line element itself we can have various variations like linear variation, quadratic variation, cubic variations like that a series of elements we can have. So, very similar way when we have discretized in 2D we can have triangle elements which is symbol most elements and then triangle also we can have cubic variation, quadratic variation and then we can go over rectangle elements. So, then hexagonal elements a number of type of elements are available in finite element method. So, that is the beauty of the finite element method and then as I mentioned the time is considered you can either. So, we can directly utilize the finite element scheme for the time also, but generally we prefer to use the finite difference schemes since otherwise the method become more complex. So, the some of the formulations which we have discussed here we have utilized the finite element in space and finite difference in time.