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Finite Element Method - Generalization of the finite element concents galerkin - weighted residual and variational approaches _03 The description of the laws of physics for space- and time-dependent problems are usually expressed in terms of partial differential equations (PDEs). For the vast majority of geometries and problems, these PDEs cannot be solved with analytical methods. Instead, an approximation of the equations can be constructed, typically based upon different types of discretizations. These discretization methods approximate the PDEs with numerical model equations, which can be solved using numerical methods. The solution to the numerical model equations are, in turn, an approximation of the real solution to the PDEs. The finite element method (FEM) is used to compute such approximations.
3 Generalization of the finite element concepts Galerkin-weighted residual and variational approaches 3.1 Introduction We have so far dealt with one possible approach to the approximate solution of the particular problem of linear elasticity Many other continuum problems arise in engineering and physics and usually these problems are posed by appropriate differential equations and boundary conditions to be imposed on the unknown function or functions It is the object of this chapter to show that all such problems can be dealt with by the finite element method Posing the problem to be solved in its most general terms we find that we seek an unknown function u such that it satisfies a certain differential equation set in a ‘domain’ (volume, area, etc.) R (Fig 3.1), together with certain boundary conditions B(u) = { :;} =o (3.2) on the boundaries r of the domain (Fig 3.1) The function sought may be a scalar quantity or may represent a vector of several variables Similarly, the differential equation may be a single one or a set of simultaneous equations and does not need to be linear It is for this reason that we have resorted to matrix notation in the above The finite element process, being one of approximation, will seek the solution in the approximate form MU Niai = Na = i= (3.3) 40 Generalization of the finite element concepts Fig 3.1 Problem domain R and boundary r where Ni are shape functions prescribed in terms of independent variables (such as the coordinates x, y , etc.) and all or most of the parameters are unknown We have seen that precisely the same form of approximation was used in the displacement approach to elasticity problems in the previous chapter We also noted there that ( a ) the shape functions were usually defined locally for elements or subdomains and (b) the properties of discrete systems were recovered if the approximating eqations were cast in an integral form [viz Eqs (2.22)-(2.26)] With this object in mind we shall seek to cast the equation from which the unknown parameters are to be obtained in the integral form (3.4) in which Gj and gj prescribe known functions or operators These integral forms will permit the approximation to be obtained element by element and an assembly to be achieved by the use of the procedures developed for standard discrete systems in Chapter 1, since, providing the functions Gj and gj are integrable, we have where W is the domain of each element and reits part of the boundary Two distinct procedures are available for obtaining the approximation in such integral forms The first is the method of weighted residuals (known alternatively as the Galerkin procedure); the second is the determination of variational functionals for which stationarity is sought We shall deal with both approaches in turn If the differential equations are linear, Le., if we can write (3.1) and (3.2) as = Lu + p = in (3.6) B(u)rMu+t=O onr (3.7) A(u) Introduction 41 then the approximating equation system (3.4) will yield a set of linear equations of the form (3.8) Ka+f=O with The reader not used to abstraction may well now be confused about the meaning of the various terms We shall introduce here some typical sets of differential equations for which we will seek solutions (and which will make the problems a little more definite) Example Steady-state heat conduction equations in a two-dimensional domain: A ( $ ) = -x:( )+-(: ): k- B(4)= 4- = or B ( $ ) = k -a4 + q = -O dn k- +Q=O on r4 (3.10) onr, where u indicates temperature, k is the conductivity, Q is a heat source, and ij are the prescribed values of temperature and heat flow on the boundaries and n is the direction normal to r In the above problem k and Q can be functions of position and, if the problem is non-linear, of or its derivatives Example Steady-state heat conduction-convection equation in two dimensions: with boundary conditions as in the first example Here u , and ~ uy are known functions of position and represent velocities of an incompressible fluid in which heat transfer occurs Example A system of three first order equations equivalent to Example 1: (3.12) 42 Generalization of the finite element concepts in R and B(u) = q5 - = on onr, =qn-q=O where qn is the flux normal to the boundary Here the unknown function vector u corresponds to the set This last example is typical of a so-called mixed formulation In such problems the number of dependent unknowns can always be reduced in the governing equations by suitable algebraic operations, still leaving a solvable problem [e.g., obtaining Eq (3.10) from (3.12) by eliminating qx and q,,] If this cannot be done [viz Eq (3.10)] we have an irreducible formulation Problems of mixed form present certain complexities in their solution which we shall discuss in Chapters 11-13 In Chapter we shall return to detailed examples of the above field problems, and other examples will be introduced throughout the book The three sets of problems will, however, be useful in their full form or reduced to one dimension (by suppressing the y variable) to illustrate the various approaches used in this chapter Weiahted residual methods 3.2 Integral or 'weak' statements equivalent to the differential equations As the set of differential equations (3.1) has to be zero at each point of the domain R, it follows that [a vTA(u) dR [qA I(u) where v={ + w2A2(u) + ]dR ") k (3.13) (3.14) is a set of arbitrary functions equal in number to the number of equations (or components of u) involved The statement is, however, more powerful We can assert that if(3.13) is satis-edfor all v then the differential equations (3.1)must be satis-ed at all points of the domain The proof of the validity of this statement is obvious if we consider the possibility that A(u) # at any point or part of the domain Immediately, a function v can be found which makes the integral of (3.13) non-zero, and hence the point is proved Integral or ‘weak’ statements equivalent to the differential equations 43 If the boundary conditions (3.12) are to be simultaneously satisfied, then we require that VTB(u)d r Jr [GIB1(u) for any set of functions V Indeed, the integral statement that la vTA(u)dR + + Z12B2(~)+ ] d r = lr VTB(u)d r = (3.15) (3.16) is satisfied for all v and V is equivalent to the satisfaction of the differential equations (3.1) and their boundary conditions (3.2) In the above discussion it was implicitly assumed that integrals such as those in Eq (3.16) are capable of being evaluated This places certain restrictions on the possible families to which the functions v or u must belong In general we shall seek to avoid functions which result in any term in the integrals becoming infinite Thus, in Eq (3.16) we generally limit the choice of v and V to bounded functions without restricting the validity of previous statements What restrictions need to be placed on the functions? The answer obviously depends on the order of differentiation implied in the equations A(u) [or B(u)] Consider, for instance, a function u which is continuous but has a discontinuous slope in the x-direction, as shown in Fig 3.2 which is identical to Fig 2.4 but is reproduced here for clarity We imagine this discontinuity to be replaced by a continuous variation in a very small distance A (a process known as ‘molification’) and study the behaviour of the derivatives It is easy to see that although the first derivative is not defined here, it has finite value and can be integrated easily but the second derivative tends to infinity This therefore presents difficulties if integrals are to be evaluated numerically by simple means, even though the integral is finite If such derivatives are multiplied by each other the integral does not exist and the function is known as non-square integrable Such a function is said to be C, continuous In a similar way it is easy to see that if nth-order derivatives occur in any term of A or B then the function has to be such that its n - derivatives are continuous (Cn-I continuity) On many occasions it is possible to perform an integration by parts on Eq (3.16) and replace it by an alternative statement of the form C ( V ) ~ D ( dR U) + E(V)TF(u)d r = (3.17) Jr In this the operators C to F usually contain lower order derivatives than those occurring in operators A or B Now a lower order of continuity is required in the choice of the u function at a price of higher continuity for v and V The statement (3.17) is now more ‘permissive’ than the original problem posed by Eqs (3 I), (3.2), or (3.16) and is called a weak form of these equations It is a somewhat surprising fact that often this weak form is more realistic physically than the original differential equation which implied an excessive ‘smoothness’ of the true solution Integral statements of the form of (3.16) and (3.17) will form the basis of finite element approximations, and we shall discuss them later in fuller detail Before doing so we shall apply the new formulation to an example 44 Generalization of the finite element concepts Fig 3.2 Differentiation of function with slope discontinuity (Co continuous) Example Weak form of the heat conduction equation - forced and natural boundary conditions Consider now the integral form of Eq (3.10) We can write the statement (3.16) as b )+-(; );: w - kx [:( k- ] + Q dxdy+ Jr, [ -1 V k-+q dr=O (3.18) noting that u and V are scalar functions and presuming that one of the boundary conditions, i.e., 4-6=0 is automatically satisfied by the choice of the functions q5 on ro Equation (3.18) can now be integrated by parts to obtain a weak form similar to Eq (3.17) We shall make use here of general formulae for such integration (Green's formulae) which we derive in Appendix G and which on many occasions will be Integral or 'weak' statements equivalent to the differential equations 45 useful, i.e We have thus in place of Eq (3.18) dv 84 dv 84 -k-+-k vQ dx a x ay ay (3.20) Noting that the derivative along the normal is given as (3.21) and, further, making onr V=-v (3.22) without loss of generality (as both functions are arbitrary), we can write Eq (3.20) as where the operator V is simply We note that (a) the variable has disappeared from the integrals taken along the boundary and that the boundary condition B($) 84 kdn + - r4 =0 on that boundary is automatically satisfied - such a condition is known as a natural boundary condition - and (b) if the choice of is restricted so as to satisfy the forced boundary conditions q5 - 6= 0, we can omit the last term of Eq (3.23) by restricting the choice of v to functions which give u = on r4 The form of Eq (3.23) is the weak form of the heat conduction statement equivalent to Eq (3.17) It admits discontinuous conductivity coefficients k and temperature which show discontinuous first derivatives, a real possibility not admitted in the differential form 46 Generalization of the finite element concepts 3.3 Approximation to integral formulations: the weighted residual Galerkin method If the unknown function u is approximated by the expansion (3.3), i.e., u M U = c N i a j = Na i= then it is clearly impossible to satisfy both the differential equation and the boundary conditions in the general case The integral statements (3.16) or (3.17) allow an approximation to be made if, in place of any function v, we put a finite set of approximate functions n n w, sa, v= j= v= w, sa, (3.24) ;= in which Sa, are arbitrary parameters and n is the number of unknowns entering the problem Inserting the above approximations into Eq (3.16) we have Sa? [ J a wTA(Na) dR + Jr wTB(Na)d r =0 and since Saj is arbitrary we have a set of equations which is sufficient to determine the parameters a, as (3.25) or, from Eq (3.17), / a C(wj)TD(Na)dR + Jr E(wj)TF(Na)d r = j = to n (3.26) If we note that A(Na) represents the residual or error obtained by substitution of the approximation into the differential equation [and B(Na), the residual of the boundary conditions], then Eq (3.25) is a weighted integral of such residuals The approximation may thus be called the method of weighted residuals In its classical sense it was first described by Crandall,' who points out the various forms used since the end of the last century More recently a very full expose of the method has been given by Finlayson.2 Clearly, almost any set of independent functions w, could be used for the purpose of weighting and, according to the choice of function, a different name can be attached to each process Thus the various common choices are: Point ~ o l l o c a t i o n wj ~ = Si, where Si is such that for x # xi; y # y j , w, = but Ja w, dR = I (unit matrix) This procedure is equivalent to simply making the residual zero at n points within the domain and integration is 'nominal' (incidentally although w, defined here does not satisfy all the criteria of Sec 3.2, it is nevertheless admissible in view of its properties) Subdornain c o l l ~ c a t i o nwj ~ = I in R, and zero elsewhere This essentially makes the integral of the error zero over the specified subdomains Approximation to integral formulations: the weighted residual Galerkin method 47 The Galerkin method (Bubnov-Galerkin).”6 wj = N j Here simply the original shape (or basis) functions are used as weighting This method, as we shall see, frequently (but by no means always) leads to symmetric matrices and for this and other reasons will be adopted in our finite element work almost exclusively The name of ‘weighted residuals’ is clearly much older than that of the ‘finite element method’ The latter uses mainly locally based (element) functions in the expansion of Eq (3.3) but the general procedures are identical As the process always leads to equations which, being of integral form, can be obtained by summation of contributions from various subdomains, we choose to embrace all weighted residual approximations under the name of generalizedfinite element method Frequently, simultaneous use of both local and ‘global’ trial functions will be found to be useful In the literature the names of Petrov and Galerkin’ are often associated with the use of weighting functions such that wj # Nj It is important to remark that the well-known finite difference method of approximation is a particular case of collocation with locally defined basis functions and is thus a case of a Petrov-Galerkin scheme We shall return to such unorthodox definitions in more detail in Chapter 16 To illustrate the procedure of weighted residual approximation and its relation to the finite element process let us consider some specific examples Example One-dimensional equation of heat conduction (Fig 3.3) The problem here will be a one-dimensional representation of the heat conduction equation [Eq (3 l o ) ] with unit conductivity (This problem could equally well represent many other physical situations, e.g., deformation of a loaded string.) Here we have d2d (3.27) A(@)=T+Q=O (O