Chapter 4. Two-Dimensional Finite Element Analysis general two-dimensional boundary-value problems 4.1 The Boundary-Value Problem 2nd-order differential equation to consider α α β φ Ω (4.1) Laplace, Poisson and Helmholtz equations φ α α β boundary conditions for φ : φ on Γ 1 : Dirichlet condition (4.2) α ˆ α ˆ ˆ γ φ Γ (4.3) third kind p, γ, q : known parameters Γ Γ Γ : Boundary of domain Ω In case α α has discontinuities and there exits no source at the discontinuity interface φ φ Γ (4.4)
α α ˆ α ˆ α ˆ ˆ ˆ ˆ Γ (4.5) 4.2 The Variational Formulation formulation of finite element equation variational problem equivalent to the boundary-value problem δ φ (4.6) φ Γ where Γ γ φ φγ φ Ω Ω α Ω α β φ (4.7) Proof of variational formulation ; Variation of φ : δ Ω α δ φ δφ α β φ δφω Γ γφδφ Γ Ω δ φω (4.8) Using identities α δφ α δ φ α δφ (4.9)
α δ φ α δ φ α δ φ (4.10) and the divergence theorem (assuming α α are continuous) Ω Ω ˆ ˆ ˆ Γ (4.11) Γ (4.8) becomes δ Ω α α β φ δ φ Ω Γ α ˆ α ˆ ˆ δ φ Γ Γ γ φ δ φ Γ boundary condition : (4.14) δ φ Ω φ Γ δ φ Γ Γ α α β φ δ φ Ω α ˆ α ˆ ˆ γ φ δ φ Γ impose stationarity requirement δ : Ω Γ α α β φ δ φ Ω α ˆ α ˆ ˆ γ φ δ φ Γ Since δ φ is arbitrary α α β φ (4.13)
(4.15) (3.1)-(3.3) formulation α ˆ α ˆ ˆ γ φ Γ (4.16) comparing (4.15) and (4.16) with (4.1) and (4.3) ; variational formulation (3.6) is equivalent to (4.2) : essential boundary condition must be imposed during the formulation (4.16) : natural boundary condition imposed automatically in the variational In case α α have discontinuity on Γ : split the entire domain into two subdomain and add terms from integration by parts (4.14) α Γ Γ α ˆ α ˆ ˆ δ φ Γ ˆ α (4.17) φ is continuous across Γ d : δφ + =δφ - Γ α α ˆ α ˆ α (4.18) since δφ is an arbitrary variation α α ˆ α ˆ α ˆ ˆ δ φ Γ ˆ ˆ ˆ ˆ δ φ Γ ˆ ˆ ˆ ˆ
(4.19) same as (4.5) and satisfied automatically as a natural condition implicitly satisfied in the process of extremization of the functional functional (4.7) is valid for both real and complex α α β γ ; for real parameters Ω α α Γ γ φ φ Γ β φ Ω (4.20) Ω φ φ Ω
4.3 Finite Element Analysis linear triangular elements 4.3.1 Discretization and Interpolation Divide the domain into triangular subdomains good discretization : avoid narrow elements : may introduce errors equilateral elements recommended identification of each element : consistent relation between element numbers and node numbers global and local node numbers, and the element numbers Introduce integer array, : local node number : element number : global node number(connectivity array) Example : 4 elements and 6 nodes(fig. 4.2) numbering : not unique, but must be consistent (clockwise or counterclockwise) easy to apply boundary conditions
Other points to consider : 1. coordinates of the nodes, 2. α α β for each element 3. p for the node at Γ 1 4. γ and q along Γ 2 4.3.2 Elemental Interpolation linear triangular elements approximation of φ within eth element : φ (4.22) for three node numbers : φ φ φ Solve for and substitute back into (4.22) : φ φ (4.23) where interpolation or expansion functions are Δ (4.24) with
and Δ Note that δ. See Fig. 4.4 4.3.3 Formulation via the Ritz method formulation of the system of equations using Ritz method or Galerkin's method A. Derivation of elemental equations using Ritz method Assume homogeneous Neumann condition : γ Functional for (4.7) : where (4.26) α Ω φ α φ β Ω Ω φ Ω (4.27) (4.27) (4.26) and take the derivative
φ Ω α α β Ω Ω (4.28) in matrix form where (4.29) φ φ Ω α α β (4.30) Ω (4.31) α α are constant or approximately constant : Ω Δ (4.32) resulting in symmetric [K e ] as Δ α α Δ β δ (4.33) Δ (4.34)
where α β α β within eth element
B. Assembly to form the System of Equations (4.29) : elemental equation system of equations : summation over all elements φ (4.35) in the expanded form ; where (4.36) (4.37) Assemble of [K] : Fig. 4.2 6 6 matrix as (4.38) add to [K] ; (4.39) in the same way add and to [K] ;
Similarly assemble [b] : (4.41) (4.42) C. Incorporation of the Boundary condition of the 3rd kind general form nonvanishing γ : add term Γ γ φ φ Γ (4.43) Assume Γ 2 is comprised by M s segments ; (4.44) where : integral over segment s.
Approximate the unknown φ within each element as where φ (4.45) ξ ξ (4.46) Differentiation of (4.43) φ γ ξ ξ (4.47) In matrix form (4.48) where γ ξ (4.49) ξ (4.50) Analytic evaluation of (4.49) and (4.50) γ δ (4.51) (4.52) Modification of (4.35) as φ φ
(4.53) augmented matrices : need an array that relates the segments and the global number of the nodes The array ns(i,s) (i=1,2; s=1,2,3,...,ms) : store the global number of the ith node of the sth segment If Γ 2 is comprised the segments on the sides defined by the nodes 6, 4, 1, 2 and 3 ; with ns(i,s) [K s ] assembled to [K] by adding each and [b s ] assembled to [b] by adding. D. Imposition of Dirichlet Boundary condition Dirichlet boundary condition at Γ 1 : assume the nodes 3, 5, and 6 are on Γ 1 with prescribed values. Impose the condition φ as (4.55) (4.55) destroys the symmetry of [K] : make modification as (4.56) In the similar manner, impose the conditions
φ φ, and the resulting matrix (4.57) the [b] becomes (4.58) (4.59) Delete 3rd, 5th and 6th equation : φ φ φ For a general problem ; Γ global node numbers in a vector, prescribed values of φ in a vector, (4.60) (4.61) for Another approach to impose Dirichlet conditions :
The case of φ ; instead of (4.55), assign a very large number, say : set (4.62) The equation associated with φ : φ φ φ φ φ φ (4.63) In a similar manner, impose φ φ : φ φ φ (4.64) note that the symmetry is retained. For a general problem having Γ this approach the boundary conditions by setting (4.65) for Exercise 4.3 Poisson's equation A φ φ B C D φ φ.
homogeneous Neumann boundary conditions at AC and BD. 4.3.4 Formulation via Galerkin's Method The residual of (4.1) : α α β φ (4.66) the weighted residual for element e : (4.66) (4.67) (4.67) Ω Ω α α β φ Using vector identities and divergence theorem (4.68) α Ω α Ω Γ Γ β φ (4.69) where Γ e : contour enclosing Ω e : outward unit vector normal to Γ e and α α Substitution of
φ φ leads to the elemental equation ; or Ω α Ω α Γ Γ β (4.70) (4.71) where Γ (4.72) Γ System of equations; or (4.74) assembly of [g]: φ (4.73) (4.75) Calculation of : Γ Γ (4.72)
Note that at the element side opposite to node i. and Γ (4.76) Γ Γ (4.77) Γ Since at the element side connecting nodes 2 and 4. Assumed is continuous within Ω. Similarly between nodes 2 and 5 ; and Γ Γ Γ Γ (4.78) Γ (4.79) Γ (4.80) Γ (4.81) Γ (4.82) The internal element sides do not contribute [g]. Generally, Γ Γ Γ Γ
(4.83), (See Fig. 4.5) If the node i is residing on Γ 1 and Dirichlet B.C. applies discard g i. If the node i is residing on the boundary where the homogeneous Neumann B.C. applies, g i =0 If the node i is residing on Γ 1 and the third kind B.C. applies ; γ φ Γ Γ Γ γ φ Γ (4.84) Since is linear function, ξγφ ξ or ξγ φ ξ (4.85) γ ξ γ φ ξ (4.86) or φ (4.87) 4.3.5 Solution of the system of equations final step : solve φ [K] : symmetric tridiagonal matrix Gauss elimination is effective