Problem 1: Consider the steady state form of the convection diffusion equation with concentra- tion, C(x), being a spatially dependent variable: dC +(dc)++ (VC) + GC = 0, dx dx dx x = [0, 10] Assume that the velocity V, diffusion D and decay/growth G parameters are spatially constant. Boundary conditions are C|x=0 = 5 [DC] dx x=10 = 3 a) Develop the fundamental and symmetrical weak forms of this equation. What is the Sobelov Space requirement for the fundamental weak form? What is the Sobelov Space requirement for the symmetrical weak form? What interpolation functions would you use in a finite ele- ment solution for each form? b) Map the elemental system of equations onto a unit coordinate system for each of the three matrices in this problem. Describe your steps. Why is this useful? c) Assume you are using Lagrange quadratic elements, each with three nodes. Develop the elemental advection, diffusion, and growth/decay matrices as well as the natural boundary condition vector. d) Assemble the elemental matrices into a global system. Also assemble the natural boundary condition vector. Assume you are considering two 3 node quadratic elements. e) Enforce the essential boundary condition. f) Assume you run a second order accurate in space code to solve the steady state convection diffusion equation using linear finite elements. You apply four different levels of spatial resolution: Ax₁ = 2, Ax2 = 1, Ax3 = 0.5, and Ax4 = 0.25, with corresponding solutions at x=10: C₁ 12.49523, C2 = 9.50908, C3 = 8.57603, and C5 = 8.34275. Estimate errors for each mesh and determine when you are in the assymptotic range. Indicate when you think you have a reliable error estimate. Explain how you are making your estimates.
Problem 1: Consider the steady state form of the convection diffusion equation with concentra- tion, C(x), being a spatially dependent variable: dC +(dc)++ (VC) + GC = 0, dx dx dx x = [0, 10] Assume that the velocity V, diffusion D and decay/growth G parameters are spatially constant. Boundary conditions are C|x=0 = 5 [DC] dx x=10 = 3 a) Develop the fundamental and symmetrical weak forms of this equation. What is the Sobelov Space requirement for the fundamental weak form? What is the Sobelov Space requirement for the symmetrical weak form? What interpolation functions would you use in a finite ele- ment solution for each form? b) Map the elemental system of equations onto a unit coordinate system for each of the three matrices in this problem. Describe your steps. Why is this useful? c) Assume you are using Lagrange quadratic elements, each with three nodes. Develop the elemental advection, diffusion, and growth/decay matrices as well as the natural boundary condition vector. d) Assemble the elemental matrices into a global system. Also assemble the natural boundary condition vector. Assume you are considering two 3 node quadratic elements. e) Enforce the essential boundary condition. f) Assume you run a second order accurate in space code to solve the steady state convection diffusion equation using linear finite elements. You apply four different levels of spatial resolution: Ax₁ = 2, Ax2 = 1, Ax3 = 0.5, and Ax4 = 0.25, with corresponding solutions at x=10: C₁ 12.49523, C2 = 9.50908, C3 = 8.57603, and C5 = 8.34275. Estimate errors for each mesh and determine when you are in the assymptotic range. Indicate when you think you have a reliable error estimate. Explain how you are making your estimates.
Chapter2: Loads On Structures
Section: Chapter Questions
Problem 1P
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Solve the question on the finite element method in Engineering; refer to the screenshot
![Problem 1: Consider the steady state form of the convection diffusion equation with concentra-
tion, C(x), being a spatially dependent variable:
dC
+(dc)++ (VC) + GC = 0,
dx
dx
dx
x = [0, 10]
Assume that the velocity V, diffusion D and decay/growth G parameters are spatially constant.
Boundary conditions are
C|x=0 = 5
[DC]
dx x=10
= 3
a) Develop the fundamental and symmetrical weak forms of this equation. What is the Sobelov
Space requirement for the fundamental weak form? What is the Sobelov Space requirement
for the symmetrical weak form? What interpolation functions would you use in a finite ele-
ment solution for each form?
b) Map the elemental system of equations onto a unit coordinate system for each of the three
matrices in this problem. Describe your steps. Why is this useful?
c) Assume you are using Lagrange quadratic elements, each with three nodes. Develop the
elemental advection, diffusion, and growth/decay matrices as well as the natural boundary
condition vector.
d) Assemble the elemental matrices into a global system. Also assemble the natural boundary
condition vector. Assume you are considering two 3 node quadratic elements.
e) Enforce the essential boundary condition.
f)
Assume you run a second order accurate in space code to solve the steady state
convection diffusion equation using linear finite elements. You apply four different levels of spatial
resolution: Ax₁ = 2, Ax2 = 1, Ax3 = 0.5, and Ax4 = 0.25, with corresponding solutions at x=10:
C₁
12.49523, C2 = 9.50908, C3 = 8.57603, and C5 = 8.34275. Estimate errors for each mesh
and determine when you are in the assymptotic range. Indicate when you think you have a reliable
error estimate. Explain how you are making your estimates.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6e098a1b-6d5d-48d5-a6ed-26facdf391a7%2F66c969a8-8dec-4695-b0ee-185fd15bc5ae%2Fewc7j1i_processed.png&w=3840&q=75)
Transcribed Image Text:Problem 1: Consider the steady state form of the convection diffusion equation with concentra-
tion, C(x), being a spatially dependent variable:
dC
+(dc)++ (VC) + GC = 0,
dx
dx
dx
x = [0, 10]
Assume that the velocity V, diffusion D and decay/growth G parameters are spatially constant.
Boundary conditions are
C|x=0 = 5
[DC]
dx x=10
= 3
a) Develop the fundamental and symmetrical weak forms of this equation. What is the Sobelov
Space requirement for the fundamental weak form? What is the Sobelov Space requirement
for the symmetrical weak form? What interpolation functions would you use in a finite ele-
ment solution for each form?
b) Map the elemental system of equations onto a unit coordinate system for each of the three
matrices in this problem. Describe your steps. Why is this useful?
c) Assume you are using Lagrange quadratic elements, each with three nodes. Develop the
elemental advection, diffusion, and growth/decay matrices as well as the natural boundary
condition vector.
d) Assemble the elemental matrices into a global system. Also assemble the natural boundary
condition vector. Assume you are considering two 3 node quadratic elements.
e) Enforce the essential boundary condition.
f)
Assume you run a second order accurate in space code to solve the steady state
convection diffusion equation using linear finite elements. You apply four different levels of spatial
resolution: Ax₁ = 2, Ax2 = 1, Ax3 = 0.5, and Ax4 = 0.25, with corresponding solutions at x=10:
C₁
12.49523, C2 = 9.50908, C3 = 8.57603, and C5 = 8.34275. Estimate errors for each mesh
and determine when you are in the assymptotic range. Indicate when you think you have a reliable
error estimate. Explain how you are making your estimates.
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