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Lecture No. 7 Finite Element Method • Define the Approximating Functions Locally Over “Finite Elements” Advantages • It

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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.
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  • Finite Element Methods

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    Part I Finite Element Methods Chapter 1 Approximation of Functions 1.1 Approximation of Functions Many successful numerical methods for differential equations aim at approx- imating the unknown function by a sum N u(x)= ciϕi(x), (1.1) i=0 X where ϕi(x) are prescribed functions and ci, i = 0,...,N, are unknown coffi- cients to be determined. Solution methods for differential equations utilizing (1.1) must have a principle for constructing N + 1 equations to determine c0,...,cN . Then there is a machinery regarding the actual constructions of the equations for c0,...,cN in a particular problem. Finally, there is a solve phase for computing solution c0,...,cN of the N + 1 equations. Especially in the finite element method, the machinery for constructing the equations is quite comprehensive, with many mathematical and imple- mentational details entering the scene at the same time. From a pedagogical point of view it can therefore be wise to introduce the computational ma- chinery for a trivial equation, namely u = f. Solving this equation with f given and u on the form (1.1) means that we seek an approximation u to f. This approximation problem has the advantage of introducing most of the finite element toolbox, but with postponing variational forms, integration by parts, boundary conditions, and coordinate mappings. It is therefore from a pedagogical point of view advantageous to become familiar with finite ele- ment approximation before addressing finite element methods for differential equations. First, we refresh some linear algebra concepts about approximating vec- tors in vector spaces. Second, we extend these concepts to approximating functions in function spaces, using the same principles and the same nota- tion.
  • Basis Functions

    Basis Functions

    Basis Functions Volker Tresp Summer 2015 1 I am an AI optimist. We've got a lot of work in machine learning, which is sort of the polite term for AI nowadays because it got so broad that it's not that well defined. Bill Gates (Scientific American Interview, 2004) \If you invent a breakthrough in artificial intelligence, so machines can learn," Mr. Gates responded,\that is worth 10 Microsofts."(Quoted in NY Times, Monday March 3, 2004) 2 Nonlinear Mappings and Nonlinear Classifiers • Regression: { Linearity is often a good assumption when many inputs influence the output { Some natural laws are (approximately) linear F = ma { But in general, it is rather unlikely that a true function is linear • Classification: { Similarly, it is often not reasonable to assume that the classification boundaries are linear hyper planes 3 Trick • We simply transform the input into a high-dimensional space where the regressi- on/classification is again linear! • Other view: let's define appropriate features • Other view: let's define appropriate basis functions 4 XOR is not linearly separable 5 Trick: Let's Add Basis Functions • Linear Model: input vector: 1; x1; x2 • Let's consider x1x2 in addition • The interaction term x1x2 couples two inputs nonlinearly 6 With a Third Input z3 = x1x2 the XOR Becomes Linearly Separable f(x) = 1 − 2x1 − 2x2 + 4x1x2 = φ1(x) − 2φ2(x) − 2φ3(x) + 4φ4(x) with φ1(x) = 1; φ2(x) = x1; φ3(x) = x2; φ4(x) = x1x2 7 f(x) = 1 − 2x1 − 2x2 + 4x1x2 8 Separating Planes 9 A Nonlinear Function 10 f(x) = x − 0:3x3 2 3 Basis functions φ1(x) = 1; φ2(x)
  • Choosing Basis Functions and Shape Parameters for Radial Basis Function Methods

    Choosing Basis Functions and Shape Parameters for Radial Basis Function Methods

    Choosing Basis Functions and Shape Parameters for Radial Basis Function Methods Michael Mongillo October 25, 2011 Abstract Radial basis function (RBF) methods have broad applications in numerical analysis and statistics. They have found uses in the numerical solution of PDEs, data mining, machine learning, and kriging methods in statistics. This work examines the use of radial basis func- tions in scattered data approximation. In particular, the experiments in this paper test the properties of shape parameters in RBF methods, as well as methods for finding an optimal shape parameter. Locating an optimal shape parameter is a difficult problem and a topic of current research. Some experiments also consider whether the same methods can be applied to the more general problem of selecting basis functions. 1 Introduction Radial basis function (RBF) methods are an active area of mathematical research. The initial motivation for RBF methods came from geodesy, mapping, and meteorology. They have since found applications in other areas, such as the numerical solution of PDEs, machine learning, and statistics. In the 1970s, Rolland Hardy suggested what he called the multiquadric method for ap- plications in cartography because he was not satisfied with the results of polynomial interpolation. His new method for fitting data could handle unevenly distributed data sites with greater consis- tency than previous methods [Har90]. The method of RBF interpolation used in this paper is a generalization of Hardy’s multiquadric and inverse multiquadric methods. Many RBFs are defined by a constant called the shape parameter. The choice of basis function and shape parameter have a significant impact on the accuracy of an RBF method.