Homework Help is Here – Start Your Trial Now!
Business
Operations Management
We would like to calculate the Observed Order of Accuracy (p) of our analysis. Consider the displacement (extension) as the System Response Quantity (SRQ) of this simulation. SRQ is a field variable that we would like to evaluate our simulation based on that. We have the exact solution for the extension so it's possible to calculate the exact value of the nodal displacement at any x position and calculate the error which is the difference of the obtained value from the simulation and the exact mathematical solution of the SQR. Since we have several nodes, to define an overall error of the SRQ, we have to consider all nodal errors. Therefore, we would like to define an error equation with the concept of average/mean. We use Linorm which provides a measure of the average absolute error over the domain and can be defined for uniform meshes as ||u - Urer || 1 =n=1|un - Uref.nl, (1) where the subscript n refers to a summation over all N nodes of our numerical domain, and un, is the numerical nodal solution and urefn is the reference solution (exact solution) of node n. Keep in mind that u is a vector so the norm of the vector is shown in || ||. If h is the element size (discretization parameter), 2h and h represent the element size of coarse and fine grid meshes respectively¹. Note that in our exam, the ratio of coarse to fine grid mesh spacing is 2 which in general is shown as r hcourse hfine = 2. (2)

We would like to calculate the Observed Order of Accuracy (p) of our analysis. Consider the displacement (extension) as the System Response Quantity (SRQ) of this simulation. SRQ is a field variable that we would like to evaluate our simulation based on that. We have the exact solution for the extension so it's possible to calculate the exact value of the nodal displacement at any x position and calculate the error which is the difference of the obtained value from the simulation and the exact mathematical solution of the SQR. Since we have several nodes, to define an overall error of the SRQ, we have to consider all nodal errors. Therefore, we would like to define an error equation with the concept of average/mean. We use Linorm which provides a measure of the average absolute error over the domain and can be defined for uniform meshes as ||u - Urer || 1 =n=1|un - Uref.nl, (1) where the subscript n refers to a summation over all N nodes of our numerical domain, and un, is the numerical nodal solution and urefn is the reference solution (exact solution) of node n. Keep in mind that u is a vector so the norm of the vector is shown in || ||. If h is the element size (discretization parameter), 2h and h represent the element size of coarse and fine grid meshes respectively¹. Note that in our exam, the ratio of coarse to fine grid mesh spacing is 2 which in general is shown as r hcourse hfine = 2. (2)

Practical Management Science
Practical Management Science
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
Chapter11: Simulation Models
Section11.4: Marketing Models
Problem 35P
See similar textbooks
icon
Related questions
Question
We would like to calculate the Observed Order of Accuracy (p) of our analysis. Consider the displacement (extension) as the System Response Quantity (SRQ) of this simulation. SRQ is a field variable that we would like to evaluate our simulation based on that. We have the exact solution for the extension so it's possible to calculate the exact value of the nodal displacement at any x position and calculate the error which is the difference of the obtained value from the simulation and the exact mathematical solution of the SQR. Since we have several nodes, to define an overall error of the SRQ, we have to consider all nodal errors. Therefore, we would like to define an error equation with the concept of average/mean. We use L₁norm which provides a measure of the average absolute error over the domain and can be defined for uniform meshes as ||u - Uref || 1 = = =1|un - Uref.nl, where the subscript n refers to a summation over all N nodes of our numerical domain, and un is the numerical nodal solution and urefn is the reference solution (exact solution) of node n. Keep in mind that u is a vector so the norm of the vector is shown in || ||. If h is the element size (discretization parameter), 2h and h represent the element size of coarse and fine grid meshes respectively. Note that in our exam, the ratio of coarse to fine grid mesh spacing is 2 which in general is shown as r= We define the error as hcourse hfine p = = 2. ¹ Oberkampf, William L., and Christopher J. Roy. Verification and validation in scientific computing. Cambridge University Press, 2010. (1) In(²2h) inten. In 2 (2) ε = U-Uref, (3) where for the fine mesh is called as & and for the coarse mesh would be E2h. In our analysis we would like to use the L₁norm definition for & to calculate the observed order of accuracy of the whole domain. The observed order of accuracy for two coarse and fine grid meshes is defined as: (4) Obtain p for the meshes of 5 and 10 elements, and also 10 and 20 elements of Problem 2.
Transcribed Image Text:We would like to calculate the Observed Order of Accuracy (p) of our analysis. Consider the displacement (extension) as the System Response Quantity (SRQ) of this simulation. SRQ is a field variable that we would like to evaluate our simulation based on that. We have the exact solution for the extension so it's possible to calculate the exact value of the nodal displacement at any x position and calculate the error which is the difference of the obtained value from the simulation and the exact mathematical solution of the SQR. Since we have several nodes, to define an overall error of the SRQ, we have to consider all nodal errors. Therefore, we would like to define an error equation with the concept of average/mean. We use L₁norm which provides a measure of the average absolute error over the domain and can be defined for uniform meshes as ||u - Uref || 1 = = =1|un - Uref.nl, where the subscript n refers to a summation over all N nodes of our numerical domain, and un is the numerical nodal solution and urefn is the reference solution (exact solution) of node n. Keep in mind that u is a vector so the norm of the vector is shown in || ||. If h is the element size (discretization parameter), 2h and h represent the element size of coarse and fine grid meshes respectively. Note that in our exam, the ratio of coarse to fine grid mesh spacing is 2 which in general is shown as r= We define the error as hcourse hfine p = = 2. ¹ Oberkampf, William L., and Christopher J. Roy. Verification and validation in scientific computing. Cambridge University Press, 2010. (1) In(²2h) inten. In 2 (2) ε = U-Uref, (3) where for the fine mesh is called as & and for the coarse mesh would be E2h. In our analysis we would like to use the L₁norm definition for & to calculate the observed order of accuracy of the whole domain. The observed order of accuracy for two coarse and fine grid meshes is defined as: (4) Obtain p for the meshes of 5 and 10 elements, and also 10 and 20 elements of Problem 2.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 3 steps with 8 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Knowledge Booster
Automation of Manufacturing Processes
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, operations-management and related others by exploring similar questions and additional content below.
Similar questions
  • SEE MORE QUESTIONS
Recommended textbooks for you
Practical Management Science
Practical Management Science
Operations Management
ISBN:
9781337406659
Author:
WINSTON, Wayne L.
Publisher:
Cengage,
SEE MORE TEXTBOOKS