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
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
Chapter2: Introduction To Spreadsheet Modeling
Section: Chapter Questions
Problem 20P: Julie James is opening a lemonade stand. She believes the fixed cost per week of running the stand...
Related questions
Question
![### Analyzing Observed Order of Accuracy
In our analysis, we aim to calculate the **Observed Order of Accuracy (\( \hat{p} \))**. We'll focus on displacement as the **System Response Quantity (SRQ)**. SRQ is crucial for evaluating our simulation, and since we have the exact solution for displacement, we can measure the error as the difference between the simulation results and exact solutions.
For comprehensive analysis, we need to consider the errors across all nodes. We employ the **\( L_1 \) norm** to measure average absolute error over the domain:
\[
\| u - u_{\text{ref}} \|_1 = \frac{1}{N} \sum_{n=1}^{N} | u_n - u_{\text{ref},n} |, \tag{1}
\]
where \( N \) is the total number of nodes, \( u_n \) is the numerical solution at node \( n \), and \( u_{\text{ref},n} \) is the exact solution. \( u \) is a vector, and its norm is indicated accordingly. The parameter \( h \) denotes element size, with \( 2h \) and \( h \) representing coarse and fine grid elements. The grid spacing ratio is \( r = \frac{h_{\text{coarse}}}{h_{\text{fine}}} = 2 \).
### Error Definition
Error is defined as:
\[
\varepsilon = u - u_{\text{ref}}, \tag{3}
\]
where \( \varepsilon_h \) is used for the fine mesh and \( \varepsilon_{2h} \) for the coarse mesh.
The observed order of accuracy (\( \hat{p} \)) for two meshes is calculated using:
\[
\hat{p} = \frac{\ln(\frac{\varepsilon_{2h}}{\varepsilon_h})}{\ln \, 2}. \tag{4}
\]
### Practical Application
Calculate \( \hat{p} \) for meshes with 5 and 10 elements, as well as for 10 and 20 elements from Problem 2.
### Reference
1. Oberkampf, W. L., & Roy, C. J. *Verification and validation in scientific computing*. Cambridge University Press, 2010.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4b8502c6-77e6-4ad1-93a4-15893327988c%2Fad8119b2-70f1-45fc-93c7-49fe3cf3490a%2Fq3346w9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Analyzing Observed Order of Accuracy
In our analysis, we aim to calculate the **Observed Order of Accuracy (\( \hat{p} \))**. We'll focus on displacement as the **System Response Quantity (SRQ)**. SRQ is crucial for evaluating our simulation, and since we have the exact solution for displacement, we can measure the error as the difference between the simulation results and exact solutions.
For comprehensive analysis, we need to consider the errors across all nodes. We employ the **\( L_1 \) norm** to measure average absolute error over the domain:
\[
\| u - u_{\text{ref}} \|_1 = \frac{1}{N} \sum_{n=1}^{N} | u_n - u_{\text{ref},n} |, \tag{1}
\]
where \( N \) is the total number of nodes, \( u_n \) is the numerical solution at node \( n \), and \( u_{\text{ref},n} \) is the exact solution. \( u \) is a vector, and its norm is indicated accordingly. The parameter \( h \) denotes element size, with \( 2h \) and \( h \) representing coarse and fine grid elements. The grid spacing ratio is \( r = \frac{h_{\text{coarse}}}{h_{\text{fine}}} = 2 \).
### Error Definition
Error is defined as:
\[
\varepsilon = u - u_{\text{ref}}, \tag{3}
\]
where \( \varepsilon_h \) is used for the fine mesh and \( \varepsilon_{2h} \) for the coarse mesh.
The observed order of accuracy (\( \hat{p} \)) for two meshes is calculated using:
\[
\hat{p} = \frac{\ln(\frac{\varepsilon_{2h}}{\varepsilon_h})}{\ln \, 2}. \tag{4}
\]
### Practical Application
Calculate \( \hat{p} \) for meshes with 5 and 10 elements, as well as for 10 and 20 elements from Problem 2.
### Reference
1. Oberkampf, W. L., & Roy, C. J. *Verification and validation in scientific computing*. Cambridge University Press, 2010.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 8 images
Knowledge Booster
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.Recommended textbooks for you
Practical Management Science
Operations Management
ISBN:
9781337406659
Author:
WINSTON, Wayne L.
Publisher:
Cengage,
Operations Management
Operations Management
ISBN:
9781259667473
Author:
William J Stevenson
Publisher:
McGraw-Hill Education
Operations and Supply Chain Management (Mcgraw-hi…
Operations Management
ISBN:
9781259666100
Author:
F. Robert Jacobs, Richard B Chase
Publisher:
McGraw-Hill Education
Practical Management Science
Operations Management
ISBN:
9781337406659
Author:
WINSTON, Wayne L.
Publisher:
Cengage,
Operations Management
Operations Management
ISBN:
9781259667473
Author:
William J Stevenson
Publisher:
McGraw-Hill Education
Operations and Supply Chain Management (Mcgraw-hi…
Operations Management
ISBN:
9781259666100
Author:
F. Robert Jacobs, Richard B Chase
Publisher:
McGraw-Hill Education
Purchasing and Supply Chain Management
Operations Management
ISBN:
9781285869681
Author:
Robert M. Monczka, Robert B. Handfield, Larry C. Giunipero, James L. Patterson
Publisher:
Cengage Learning
Production and Operations Analysis, Seventh Editi…
Operations Management
ISBN:
9781478623069
Author:
Steven Nahmias, Tava Lennon Olsen
Publisher:
Waveland Press, Inc.