Model Solutions (week 4)

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1 CIV-E16 (17) Engineering Computation and Simuation 1 Home Exercise 6.3 Mode Soutions (week 4) Construct the inear Lagrange basis functions (noda vaues as degrees of freedom) of the ine segment reference eement with end points 1 and 1 as nodes. Construct the quadratic Lagrange basis functions of the ine segment reference eement (noda vaues as degrees of freedom) with end points 1 and 1 and nodes 1, and 1. The inear Lagrange basis functions of the ine segment reference eement can be written in the foowing genera form: The coordinates of eement nodes are N p (ξ, η) a p + b p ξ, p 1,. (.1) ξ 1 1, ξ 1,. (.) Substituting coordinate vaues (.) into the formua (.1) and using the foowing conditions N p (ξ q ) δ pq, p, q 1,, (.3) one can construct the biinear Lagrange basis functions N 1 (ξ) 1 (1 ξ), N (ξ) 1 (1 + ξ). The quadratic Lagrange basis functions of the ine segment reference eement can be written in the foowing genera form: N p (ξ) a p + b p ξ + c p ξ, p 1,, 3. (.4) The node coordinates of the reference eement are ξ 1 1, ξ, ξ 3 1. (.5) Then substituting coordinate vaues (.5) into the formua (.4) and using the foowing conditions N p (ξ q ) δ pq, p, q 1,, 3 (.6) one can construct the quadratic Lagrange basis functions N 1 (ξ) 1 ξ(1 ξ), N (ξ) 1 ξ, N 3 (ξ) 1 ξ(1 + ξ). (.7)

2 CIV-E16 (17) Engineering Computation and Simuation Home Exercise 6.4 Let us consider the beam bending probem of Home exercise 4.5. Fig..1: Heat diffusion in a square domain. Cacuate the oca eement stiffness matrix and oca eement force vector of the corresponding finite eement probem by using the reference eement approach. Assembe the corresponding goba system matrix and goba force vector composed of contributions from two eements. Weak form formuation (Home Exercise 4. ): Find M such that it satisfies M() M, M(L) M, (M ) dx <, (.8) for a v satisfying M v dx fvdx (.9) v(), v(l), (v ) dx <. (.1) First of a one shoud divide the domain (, L) into n finite eements and get the coordinates x i of n + 1 nodes, where i,..., n. After that one can construct n + 1 inear basis functions φ i (x) a i + b i x, with the foowing properties φ i (x j ) δ ij, i, j,..., n. Secondy, one can seect functions M(x) and v(x) in the foowing form M(x) n d i φ i (x), v(x) i n c i φ i (x). i

3 CIV-E16 (17) Engineering Computation and Simuation 3 Then substituting M(x) and v(x) into the equation (.9), one can obtain eements of stiffness matrix and force vector, respectivey, as K ij f i φ i(x)φ j(x)dx, i, j,..., n (.11) f(x)φ i (x)dx, i,..., n. (.1) Making the simiar operations as in the ecture notes one can obtain the formua for the eement stiffness matrix and force vector respectivey K pq x f p x 1 (ϕ p(x)) (ϕ q(x)) dx, p, q 1, (.13) x x 1 f(x)ϕ p(x)dx, p 1,, (.14) where 1,..., n. The eement stiffness matrix and force vector can be respectivey written as [ ] [ ] K K 11 K1 f K1 K, f 1 f. Then by accompishing a change of variabes x F e (ξ) 1 (x 1 + x + (x x 1 )ξ) for the integration (.13) and (.14) one can get K pq f p N p(ξ)n q(ξ)(x (ξ)) 1 dξ, p, q 1, (.15) f(x(ξ))n p (ξ)x (ξ)dξ, p 1,. (.16) Taking from the ecture notes the oca basis functions N 1 (ξ) 1 (1 ξ), N (ξ) 1 (1 + ξ) and substituting it into the formuae (.15) and (.16) one can get the oca eement stiffness matrix and oca eement force vector [ ] [ ] K 1 1 1, f fh 1, (.17) h for constant f and h x x 1.

4 CIV-E16 (17) Engineering Computation and Simuation 4 The goba stiffness matrix and goba force vector composed of contributions from three eements can be written (without taking into account the boundary conditions) as foows: K11 1 K1 1 f1 1 K K1 1 K 1 + K11 K1, f f 1 + f1. (.18) K 1 K f Home Exercise 6.5 Show that an affine mapping with a transation vector b and inear transformation matrix A. Associate each of the given transformation matrices A either to pure scaing, rotation, refection or to shearing. show that, affine mapping maps the vertices (ξ, η) x x x 3 x 1 ξ y y1 + y y1 y3 y1. (.19) η from reference triange to the actua triange (ξ, η) (, ) [ ] x x 3 x 1 x 1 y1 y1 + y y1 y3 y1 y1 ; (.) (ξ, η) (1, ) [ ] x x x 3 x 1 + y y1 y3 y1 ; (.1) y y 1 (ξ, η) (, 1) [ ] x 3 x x 3 x 1 x 3 + y y1 y3 y1 1 y3. (.) y 3 y 1 By using different types of inear transformation matrix A we observe the foowing pure (i.e. nu transation b ) y

CIV-E16 (17) Engineering Computation and Simuation 5 shear scaing rifection x 1 b ξ ξ + bη ; y 1 η η (.3) x a ξ aξ ; y d η dη (.4) x 1 ξ ξ ; y 1 η η (.5) rotation x cos(θ) sin(θ) ξ ξ cos(θ) η sin(θ).

5 CIV-E16 (17) Engineering Computation and Simuation 5 shear scaing rifection x 1 b ξ ξ + bη ; y 1 η η (.3) x a ξ aξ ; y d η dη (.4) x 1 ξ ξ ; y 1 η η (.5) rotation x cos(θ) sin(θ) ξ ξ cos(θ) η sin(θ). (.6) y sin(θ) cos(θ) η ξ sin(θ) + η cos(θ) Home Exercise 7.3 Derive the strong form, biinear form, oad functiona and the variationa space of the weak form corresponding to the Euer-Bernoui beam Fig..: Heat diffusion in a square domain. One can start from the principe of virtua work δw int δw ext. (.7) According to the ecture notes, δw int can be rewritten as foows δw int σ : δεdv and δw ext has the foowing form δw ext V V b δudv + Mδw dx, (.8) S t t δuds, (.9)

6 CIV-E16 (17) Engineering Computation and Simuation 6 with b b y (x, y, z)e y, t t y (y, z)e y and S t denoting the free end face. For the given oading combination, we get δw ext b y (x, y, z)δvdv + t y (y, z)δvds. (.3) V S t Utiizing the basic kinematic dimension reduction assumptions of the Euer Bernoui beam, the dispacements can be presented as foows v(x, y) w(x), u(x, y) yw (x). (.31) Substituting (.33) into the formua (.3) and using the foowing notations b(x) b y (x, y, z)dydz, F L t y (y, z)dydz, A(x) where A(x) denotes the cross-section area at point x (in fact, S t A(L)), one can get δw ext S t b(x)δwdx + F L δw(l). (.3) Then, by substituting the expressions for δw int and δw ext into the principe of virtua work one can obtain Mδw L M δw L + F L δw(l) + (M + b(x))δwdx, δw. (.33) By denoting M(x) EIw and Q(x) M, the strong form for the given beam probem can be written as foows: (EIw ) b(x), x (, L) (.34) w(), w (), M(L), Q(L) F L. (.35) The corresponding weak form can be obtained from the virtua work expressions above and now can be formuated in the foowing form: Find w W such that a(w, ŵ) (ŵ) ŵ W, where a(w, ŵ) EIw ŵ da, (.36) Ω Ω (ŵ) b(x)ŵda + F L ŵ(l), (.37) W { v H (Ω) v(), v () } H (Ω). (.38)

Introduction to PDEs and Numerical Methods Tutorial 10. Finite Element Analysis

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