Lecture no. 7 Finite Element Method Define the approximating functions locally over “finite elements” Advantages It’s much easier to satisfy the b.c.’s with local functions over local parts of the boundary than it is with global functions over the entire boundary. Splitting the domain into intervals and using lower order approximations within each element will cause the integral error to assure better accuracy on a pointwise basis. (Courant, 1920). An integral norm attempts to minimize total error over the entire domain. A low integral norm does not always mean that we have good pointwise error norm. CE 60130 FINITE ELEMENT M ETHODS - L E C T U R E 7 P a g e 1 | 32 General Steps to FEM Divide the domain into N sub-intervals or finite elements. Develop interpolating functions valid over each element. We will make use of localized coordinate systems to assure functional universality (i.e. they will be applicable to any length element at any location). We will tailor these functions such that the required degree of functional continuity can be readily enforced. Enforce functional continuity Through definition of Cardinal basis Through “global” matrix assembly Note that we use these interpolating functions in conjunction with the implementation of the desired weighted residual form (i.e. integrations etc.) as before. CE 60130 FINITE ELEMENT M ETHODS - L E C T U R E 7 P a g e 2 | 32 Lagrange Interpolation Lagrange Interpolation : pass an approximating function, g(x), exactly through the functional values at a set of interpolation points or nodes. CE 60130 FINITE ELEMENT M ETHODS - L E C T U R E 7 P a g e 3 | 32 Method 1 to deriving g(x) – Power series: ─ We are given 푓0, 푓1, 푓2 and corresponding points 푥0, 푥1, 푥2 ─ Constraints we can apply 푔(푥0 = 0) = 푓0 푔(푥1 = ℎ) = 푓1 푔(푥2 = 2ℎ) = 푓2 3 Constraints ⇒ 3 d.o.f/3 nodes = 1 d.o.f/node ⇒ Polynomial form g(x) can have 3 d.o.f ⇒ quadratic ─ General form of g(x) 푔(푥) = 푎 + 푏푥 + 푐푥2 ─ Apply constraints 푔(푥0 = 0) = 푓0 ⇒ 푎 = 푓0 2 푔(푥1 = ℎ) = 푓1 ⇒ 푎 + 푏ℎ + 푐ℎ = 푓1 2 푔(푥2 = 2ℎ) = 푓2 ⇒ 푎 + 2ℎ푏 + 4ℎ 푐 = 푓2 CE 60130 FINITE ELEMENT M ETHODS - L E C T U R E 7 P a g e 4 | 32 Solve a system of simultaneous equations. 1 0 0 푎 푓0 [1 1 1] [ 푏ℎ ]=[푓1] ⇒ 2 1 2 4 푐ℎ 푓2 푎 1 0 0 푓0 [ 푏ℎ ] = [−1.5 2 −0.5] [푓1] ⇒ 2 푐ℎ 0.5 −1 0.5 푓2 푎 = 푓0 1 3 1 푏 = (− 푓 + 2푓 − 푓 ) ℎ 2 0 1 2 2 1 1 1 푐 = ( 푓 − 푓 + 푓 ) ℎ2 2 0 1 2 2 1 1 ∴ 푔(푥) = 푓 + (−3푓 + 4푓 − 푓 )푥 + (푓 − 2푓 + 푓 )푥2 0 2ℎ 0 1 2 2ℎ2 0 1 2 Interpolating function throughout interval of interest [0,2ℎ] CE 60130 FINITE ELEMENT M ETHODS - L E C T U R E 7 P a g e 5 | 32 Let’s rewrite 푔(푥) Factor out of 푓0, 푓1 and 푓2 3푥 1 4 1 1 1 푔(푥) = 푓 (1 − 푥 + 푥2) + 푓 ( 푥 − 푥2) + 푓 (− 푥 + 푥2) 0 ⏟ 2ℎ 2 ℎ2 1 ⏟2ℎ ℎ2 2 ⏟ 2ℎ 2 ℎ2 ≡휙0(푥) ≡휙1(푥) ≡휙2(푥) ⇒ 2 푔(푥) = ∑푖=0 푓푖 ⏟휙푖 (푥 ) → Interpolating basis functions!! Each is associated with one node. Looks very much like expansions we used for w.r. methods!!! Let’s plot these functions 휙0(푥0 = 0) = 1 휙1(푥0 = ∅) = 0 휙2(푥0 = 0) 휙0(푥1 = ℎ) = 0 휙1(푥1 = ℎ) = 1 휙2(푥1 = ℎ) = 0 휙0(푥1 = 2ℎ) = 0 휙1(푥2 = 2ℎ) = 0 휙2(푥2 = 2ℎ) = 1 CE 60130 FINITE ELEMENT M ETHODS - L E C T U R E 7 P a g e 6 | 32 1 푖 = 푗 Thus 휙 (푥 ) = 훿 = ⇒ We can in fact use the constraints to derive 휙 , 휙 1 푗 푖푗 0 푖 ≠ 푗 0 1 and 휙2 Thus interpolating functions are 1 at nodes they are associated with and zero at all other nodes. At x values other than nodal values these functions vary and do not equal zero. A very important consequence of using Lagrange interpolation is that 푔(푥푖) = 푓푖 This property and the fact that we define nodes on inter-element boundaries will enable us to easily enforce functional continuity on inter-element boundaries CE 60130 FINITE ELEMENT M ETHODS - L E C T U R E 7 P a g e 7 | 32 Applying Lagrange Interpolation to develop 풖풂풑풑 Option 1 – Develop a higher order approximation which is global and based on Lagrange basis function defined over the entire domain. 푁 푢푎푝푝 = ∑ 푢푖휙푖 푖=1 where 휙푖 = globally defined Lagrange basis functions valid over the entire domain 푢푖 = the expansion coefficients and by definition these equal the function values at the nodes!! 푖 = 1, 푁 are the N “nodes” defined throughout the global domain These nodes can be equispaced or nonequispaced CE 60130 FINITE ELEMENT M ETHODS - L E C T U R E 7 P a g e 8 | 32 Boundary conditions can be readily incorporated into the expansion if they are function specified (essential type boundary conditions) ─ For example 푢(푥퐿) = 푢퐿 푢(푥푅) = 푢푅 ─ The expansion can now be written as 푁−1 푢푎푝푝 = 푢퐿휙1 + 푢푅휙푁 + ∑ 푢푖휙푖 푖=2 (since 푢1 = 푢퐿 and 푢푁 = 푢푅) ─ Thus 푢퐵 = 푢퐿휙1 + 푢푅휙푁 ─ We note that 푢퐵 satisfies admissibility conditions 푢퐵(푥퐿) = 푢퐿 ⏟휙 1 (푥 퐿 ) + 푢푅 ⏟휙 푁 (푥 퐿 ) = 푢퐿 =1 =0 푢퐵(푢푅) = 푢퐿 ⏟휙 1 (푥 푅 ) + 푢푅 ⏟휙 푁 (푥 푅 ) = 푢푅 =0 =1 CE 60130 FINITE ELEMENT M ETHODS - L E C T U R E 7 P a g e 9 | 32 ─ Also we note that the remaining 휙푖 푖 = 2, 푁 − 1 satisfy 휙푖(푥퐿) = 0 → 푖 = 2, 푁 − 1 휙푖(푥푅) = 0 Thus satisfying function specified b.c.’s (essential) for 1-D is very easy!! For natural b.c.’s it is much more difficult. The sequence of functions 휙푖 are linearly independent The coefficients in the expansion are now actually equal to the values of the function at the nodes!! The drawback of option 1 is that you obtain poor pointwise convergence as N becomes large for most problems. Another drawback is that the matrix will also be almost fully populated Typically the technique does work well for slowly varying smooth solutions. CE 60130 FINITE ELEMENT M ETHODS - L E C T U R E 7 P a g e 10 | 32 Example Solve 퐿(푢) = 푝(푥) 푥 ∈ [푥퐿, 푥푅] b.c.’s 푢(푥퐿) = 푢퐿 푢(푥푅) = 푢푅 Assume that we will apply a six term expansion 5 푢푎푝푝 = 푢퐿휙1 + 푢푅휙6 + ∑ 푢푖휙푖 푖=2 CE 60130 FINITE ELEMENT M ETHODS - L E C T U R E 7 P a g e 11 | 32 Note that 푢퐵 = 푢퐿휙1 + 푢푅휙6 푢퐵 satisfies b.c.’s as specified Note that 휙2, 휙3, 휙4 and 휙5 all satisfy the homogeneous form of the essential function specified b.c.’s CE 60130 FINITE ELEMENT M ETHODS - L E C T U R E 7 P a g e 12 | 32 Option 2 – Develop an approximation 푢푎푝푝 which is the sum of localized approximations 푀 푁푗 푗 푗 푢푎푝푝 = ∑ ∑ 푢푖 휙푖 푗=1 푖=1 푗 Where 푢푖 = expansion coefficient for element j and node i within element j 푗 휙푖 = Lagrange basis function for element j and node i 푗 Note that 휙푖 = 0 outside of element j 푗 = 1, 푀 = total number of localized domains or “finite elements” 푖 = 1, 푁푗 = total number of nodes within element j CE 60130 FINITE ELEMENT M ETHODS - L E C T U R E 7 P a g e 13 | 32 Local unknowns at nodes 1 1 1 3 3 3 3 3 3 푢1 푢3 푢3 푢1 푢2 푢3 푢4 푢5 푢6 ← 푒푙푒푚푒푛푡 푖푛푑푒푥 푢2 푢2 푢2 푢4 푢4 푢4 1 2 3 1 2 3 ← 푙표푐푎푙 푛표푑푒 푖푛푑푒푥 The boundary conditions can again be readily built into 1-D type problems for function specified conditions (essential). Even in multiple dimensions, function specified b.c.’s (essential) can be incorporated in a straight forward manner The sequence of all functions will be linearly independent. Again note that those coeficients of expansion are equal to the actual function values at the nodes CE 60130 FINITE ELEMENT M ETHODS - L E C T U R E 7 P a g e 14 | 32 To ensure 퐶0 inter-element functional continuity we must have at all inter-element boundaries: 푢푎푝푝(left of an inter-element boundary) = 푢푎푝푝 (right of an inter-element boundary) Thus in the example given Anywhere in element 1 1 1 1 1 1 1 1 푢 (휉) = 푢1휙1(휉) + 푢2휙2(휉) + 푢3휙3(휉) Note that all other Lagrange basis function from other elements are defined as zero. Anywhere in element 2 2 2 2 2 2 1 2 푢 (휉) = 푢1휙1 (휉) + 푢2휙2(휉) + 푢3휙3(휉) Element 1 evaluated at r.h.s. 1 1 1 푢 (휉3 ) = 푢3 Element 2 evaluated at l.h.s. 2 2 2 푢 (휉1 ) = 푢1 In order to enforce 푐0 functional continuity we must have 1 1 2 2 푢 (휉3 ) = 푢 (휉1 ) CE 60130 FINITE ELEMENT M ETHODS - L E C T U R E 7 P a g e 15 | 32 1 2 ∴ 푢3 = 푢1 In general we must have the expansion coefficients equal at inter-element boundaries in order to satisfy 푐0 functional continuity requirements In our example, the expansion coef.’s are related as 1 2 푢3 = 푢1 2 3 푢3 = 푢1 3 4 푢6 = 푢1 ∴ We must have expansion coef.’s at inter-element boundaries be equal There are several approaches to implement the inter-element functional continuity (i.e.