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Problem 3. (15 points) T3 finite element is defined over AABC (in physical coordinates). The vertices of this triangle...
T3 finite element is defined over AABC (in physical coordinates). The vertices of this triangle hav the follow...
T3 finite element is defined over AABC (in physical coordinates). The vertices of this triangle hav the following coordinates: A(-2,-1), B(3,2), and C(0,6). Problem 2 a) Using 1 point and 3 point integration rules, compute f(x, y)ds AABC 2x2-3xy + y2. where f(x,y) Which rule gives more accurate result? c) What is the integration error, if 3 point rule is used? (Hint: for what polynomial degree 3 point rule gives the exact result?) b)
T3 finite element is defined over...
T3 finite element is defined over AABC (in physical coordiinates). The vertices of this triangle have the followi...
T3 finite element is defined over AABC (in physical coordiinates). The vertices of this triangle have the following coordinates: A(-3, -5), B(2,-1), and C(-6, 4). f(x,y)ds SABC where f(x, y) 2x2-5y2 3xyx-y Solve Problem 3 using 3 point integration rule.
T3 finite element is defined over AABC (in physical coordiinates). The vertices of this triangle have the following coordinates: A(-3, -5), B(2,-1), and C(-6, 4).
f(x,y)ds SABC where f(x, y) 2x2-5y2 3xyx-y
Solve Problem 3 using 3 point integration rule.
O CWOIK 2. Submission deadline: 11:59pm on Friday June 7, 2019 Penalty for late submission: 20% per day T3 finite e...
O CWOIK 2. Submission deadline: 11:59pm on Friday June 7, 2019 Penalty for late submission: 20% per day T3 finite element is defined over AABC (in physical coordinates). The vertices of this triangle have the following coordinates: A(-2, -1), B(3,2), and C(0, 6). Problem 1 Calculate the partial derivatives of T3 basis functions with respect to the physical coordinates x and y. Problem 2. a) Using 1 point and 3 point integration rules, compute f(x, y)ds AABC where f(x,y) =2x2-3xy...
Function f(x, y) 2x2-3x +5y - y2 is going to be represented by T3 basis functions over AABC. Calculate the values of the degrees of freedom Ci in the linear combination that represents f(x,...
Function f(x, y) 2x2-3x +5y - y2 is going to be represented by T3 basis functions over AABC. Calculate the values of the degrees of freedom Ci in the linear combination that represents f(x,y): f(x, y)- CiN(x, y) T3 finite element is defined over ΔABC (in physical coordinates). The vertices of this triangle have the following coordinates: A(-2,-1), B(3,2), and C(0, 6) Problem 1
Function f(x, y) 2x2-3x +5y - y2 is going to be represented by T3 basis functions...
3. The pair of random variables X and Y is uniformly distributed on the interior of the triangle with the vertices whose coordinates are (0,0), (0,2), and (2,0) (i.e., the joint density is equal to a...
3. The pair of random variables X and Y is uniformly distributed on the interior of the triangle with the vertices whose coordinates are (0,0), (0,2), and (2,0) (i.e., the joint density is equal to a constant inside the triangle and zero outside). (a) (10 points) Find P(Y+X< 1). (b) (10 points) Find P(X = Y). (c) (10 points) Find P(Y > 1X = 1/2).
3. The pair of random variables X and Y is uniformly distributed on the interior...
Exercise 2: Finite element method We are interested in computing numerically the solution to a 2D...
Exercise 2: Finite element method We are interested in computing numerically the solution to a 2D Laplace equation u 0, The triangulated domain is given in the file mesh.mat on Blackboard. which contains the V × 2 nnatrix vertices storing the two coordinates of the vertices and a F × 3 matrix triangles in which each ro w J contains the indices in {1,····V) of the three vertices of the j-th triangle. a) Using for example MATLAB's triplot or trimesh...
3. (2 Points) Let Q be the quadrilateral in the ry-plane with vertices (1, 0), (4,0), (0, 1), (0,4). Consider 1 dA...
3. (2 Points) Let Q be the quadrilateral in the ry-plane with vertices (1, 0), (4,0), (0, 1), (0,4). Consider 1 dA I+y Deda (a) Evaluate the integral using the normal ry-coordinates. (b) Consider the change of coordinates r = u-uv and y= uv. What is the image of Q under this change of coordinates?bi (c) Calculate the integral using the change of coordinates from the previous part. Change of Variables When working integrals, it is wise to choose a...
Problem 3. (3 points). Determine the nodal displacements and reaction forces using the finite element direct...
Problem 3. (3 points). Determine the nodal displacements and reaction forces using the finite element direct method for the 1-D bar elements connected as shown below. Do not rename the nodes or elements when solving. Assume that the bars can only undergo translation in x (1 DOF at each node). Nodes 1 and 3 are fixed. Element 1 has Young's Modulus of 300 Pa, length of 1 m and cross-sectional area of m. Element 2 has Young's Modulus of 200...
Let G = (V, E) be a finite graph. We will use a few definitions for...
Let G = (V, E) be a finite graph. We will use a few definitions for the statement of this problem. The Tutte polynomial is defined as the polynomial in 2 variables, 2 and y, given by: Definition 1 Tg(x,y) = (x - 1)*(A)-k(E)(y - 1)*(A)+|A1-1V1 ACE where for A CE, k(A) is the number of connected components of the graph (V, A). For this problem we will need the following definition: Definition 2 (Acyclic Graph) A graph is called...
2. (1 Point) Let r-2u and y-3u. (a) Let R be the rectangle in the uv-plane defined by the points (0,0), (2,0), (2,...
2. (1 Point) Let r-2u and y-3u. (a) Let R be the rectangle in the uv-plane defined by the points (0,0), (2,0), (2,1), (0 , 1). Find the area of the image of R in the ry plane? (b) Find the area of R by computing the Jacobian of the transformation from uv-space to xy-space Change of Variables When working integrals, it is wise to choose a coordinate system that fits the problem; e.g. polar coordinates are a good choice...
T3 finite element is defined over AABC (in physical coordinates). The vertices of this triangle hav the following coordinates: A(-2,-1), B(3,2), and C(0,6). Problem 2 a) Using 1 point and 3 point integration rules, compute f(x, y)ds AABC 2x2-3xy + y2. where f(x,y) Which rule gives more accurate result? c) What is the integration error, if 3 point rule is used? (Hint: for what polynomial degree 3 point rule gives the exact result?) b)
T3 finite element is defined over...
T3 finite element is defined over AABC (in physical coordiinates). The vertices of this triangle have the following coordinates: A(-3, -5), B(2,-1), and C(-6, 4). f(x,y)ds SABC where f(x, y) 2x2-5y2 3xyx-y Solve Problem 3 using 3 point integration rule.
T3 finite element is defined over AABC (in physical coordiinates). The vertices of this triangle have the following coordinates: A(-3, -5), B(2,-1), and C(-6, 4).
f(x,y)ds SABC where f(x, y) 2x2-5y2 3xyx-y
Solve Problem 3 using 3 point integration rule.
O CWOIK 2. Submission deadline: 11:59pm on Friday June 7, 2019 Penalty for late submission: 20% per day T3 finite element is defined over AABC (in physical coordinates). The vertices of this triangle have the following coordinates: A(-2, -1), B(3,2), and C(0, 6). Problem 1 Calculate the partial derivatives of T3 basis functions with respect to the physical coordinates x and y. Problem 2. a) Using 1 point and 3 point integration rules, compute f(x, y)ds AABC where f(x,y) =2x2-3xy...
Function f(x, y) 2x2-3x +5y - y2 is going to be represented by T3 basis functions over AABC. Calculate the values of the degrees of freedom Ci in the linear combination that represents f(x,y): f(x, y)- CiN(x, y) T3 finite element is defined over ΔABC (in physical coordinates). The vertices of this triangle have the following coordinates: A(-2,-1), B(3,2), and C(0, 6) Problem 1
Function f(x, y) 2x2-3x +5y - y2 is going to be represented by T3 basis functions...
3. The pair of random variables X and Y is uniformly distributed on the interior of the triangle with the vertices whose coordinates are (0,0), (0,2), and (2,0) (i.e., the joint density is equal to a constant inside the triangle and zero outside). (a) (10 points) Find P(Y+X< 1). (b) (10 points) Find P(X = Y). (c) (10 points) Find P(Y > 1X = 1/2).
3. The pair of random variables X and Y is uniformly distributed on the interior...
Exercise 2: Finite element method We are interested in computing numerically the solution to a 2D Laplace equation u 0, The triangulated domain is given in the file mesh.mat on Blackboard. which contains the V × 2 nnatrix vertices storing the two coordinates of the vertices and a F × 3 matrix triangles in which each ro w J contains the indices in {1,····V) of the three vertices of the j-th triangle. a) Using for example MATLAB's triplot or trimesh...
3. (2 Points) Let Q be the quadrilateral in the ry-plane with vertices (1, 0), (4,0), (0, 1), (0,4). Consider 1 dA I+y Deda (a) Evaluate the integral using the normal ry-coordinates. (b) Consider the change of coordinates r = u-uv and y= uv. What is the image of Q under this change of coordinates?bi (c) Calculate the integral using the change of coordinates from the previous part. Change of Variables When working integrals, it is wise to choose a...
Problem 3. (3 points). Determine the nodal displacements and reaction forces using the finite element direct method for the 1-D bar elements connected as shown below. Do not rename the nodes or elements when solving. Assume that the bars can only undergo translation in x (1 DOF at each node). Nodes 1 and 3 are fixed. Element 1 has Young's Modulus of 300 Pa, length of 1 m and cross-sectional area of m. Element 2 has Young's Modulus of 200...
Let G = (V, E) be a finite graph. We will use a few definitions for the statement of this problem. The Tutte polynomial is defined as the polynomial in 2 variables, 2 and y, given by: Definition 1 Tg(x,y) = (x - 1)*(A)-k(E)(y - 1)*(A)+|A1-1V1 ACE where for A CE, k(A) is the number of connected components of the graph (V, A). For this problem we will need the following definition: Definition 2 (Acyclic Graph) A graph is called...
2. (1 Point) Let r-2u and y-3u. (a) Let R be the rectangle in the uv-plane defined by the points (0,0), (2,0), (2,1), (0 , 1). Find the area of the image of R in the ry plane? (b) Find the area of R by computing the Jacobian of the transformation from uv-space to xy-space Change of Variables When working integrals, it is wise to choose a coordinate system that fits the problem; e.g. polar coordinates are a good choice...
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