Board # 85 : Introducing the Galerkin Method of Weighted Residuals into an Undergraduate Elective Course in Finite Element Methods

Aneet Dharmavaram Narendranath

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Conference: 2017 ASEE Annual Conference & Exposition
Location: Columbus, Ohio
Publication Date: June 24, 2017
Start Date: June 24, 2017
End Date: June 28, 2017
Conference Session: Mechanical Engineering Division Poster Session
Tagged Division: Mechanical Engineering
Page Count: 28
DOI: 10.18260/1-2--27941
Permanent URL: https://peer.asee.org/27941
Download Count: 5714

Paper Authors

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Aneet Dharmavaram Narendranath Michigan Technological University

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Dr.Aneet Dharmavaram Narendranath is currently a Lecturer at Michigan Technological University (Michigan Tech). He received a PhD in Mechanical Engineering-Engineering mechanics in 2013. Subsequently, he worked as a visiting assistant professor at Michigan Tech from 2013-2014 and then as an Engineer at the French Nuclear Commission (CEA) in France until 2015. His research interests are mathematical modeling of fluid physics. His pedagogical interests are development of mathematically oriented coures in mechanical engineering.

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Abstract

Modern day finite element method (post 1940s-50s) as taught in undergraduate level (senior level) electives shows bifurcation from classical methods (pre 1900s) in at least its abstraction from rigorous mathematical concepts through the use of powerful software tools. However, it is beneficial for students of this powerful approximation technique, that is the FEM, to be made aware of the demarcation between classical methods and computer tool based analysis.

The overall objective is to introduce the Galerkin method of weighted residuals for linear ordinary differential equations and to extend that idea to linear, steady state problems in structural mechanics and thermal transport. Exposure to the Galerkin method allows students to connect differential equation based mathematical models to plane-problems in elasticity, lubrication theory problems in fluid dynamics and steady state thermal transport problems. Students are made aware of the concept of "global" vs "local" shape functions, "element order", "convergence" and "error".

The Poisson's equation u''(x)=f is primarily utilized to build the students' confidence in solving differential equations and applying the Galerkin method. This allows students to forge a connection between differential equations and simple linear (yet powerful) mathematical models. An incremental approach is taken by making the Poisson's equation complicated by including function, f, of increasing complexity. Students find appropriate polynomial functions for use in the Galerkin method of weighted residual for the Poisson's equation.

Finally, MATLAB and its partial differential equation toolbox, pdetool, is used to connect the Galerkin Method to classical engineering problems. How boundary conditions could have an effect of reducing a 2-D problem to a 1-D problem was explored. This exercise allowed students to be conscientious of boundary conditions and the variety and applicability thereof, as evidenced through examination and homework assignment results.

Homework assignments, examinations, end of semester design problem/project and student exit surveys are used as metrics to check efficacy of pedagogy.

ABET: a,b,e,g,i,k

Citation
Format

Dharmavaram Narendranath, A. (2017, June), Board # 85 : Introducing the Galerkin Method of Weighted Residuals into an Undergraduate Elective Course in Finite Element Methods Paper presented at 2017 ASEE Annual Conference & Exposition, Columbus, Ohio. 10.18260/1-2--27941

TY  - CPAPER
AB  - Modern day finite element method (post 1940s-50s) as taught in undergraduate level (senior level) electives shows bifurcation from classical methods (pre 1900s) in at least its abstraction from rigorous mathematical concepts through the use of powerful software tools.  However, it is beneficial for students of this powerful approximation technique, that is the FEM, to be made aware of the demarcation between classical methods and computer tool based analysis.

The overall objective is to introduce the Galerkin method of weighted residuals for linear ordinary differential equations and to extend that idea to linear, steady state problems in structural mechanics and thermal transport.  Exposure to the Galerkin method allows students to connect differential equation based mathematical models to plane-problems in elasticity, lubrication theory problems in fluid dynamics and steady state thermal transport problems. Students are made aware of the concept of &quot;global&quot; vs &quot;local&quot; shape functions, &quot;element order&quot;, &quot;convergence&quot; and &quot;error&quot;.  

The Poisson&#39;s equation u&#39;&#39;(x)=f is primarily utilized to build the students&#39; confidence in solving differential equations and applying the Galerkin method. This allows students to forge a connection between differential equations and simple linear (yet powerful) mathematical models.  An incremental approach is taken by making the Poisson&#39;s equation complicated by including function, f, of increasing complexity.  Students find appropriate polynomial functions for use in the Galerkin method of weighted residual for the Poisson&#39;s equation.

Finally, MATLAB and its partial differential equation toolbox, pdetool, is used to connect the Galerkin Method to classical engineering problems.  How boundary conditions could have an effect of reducing a 2-D problem to a 1-D problem was explored. This exercise allowed students to be conscientious of boundary conditions and the variety and applicability thereof, as evidenced through examination and homework assignment results.  

Homework assignments, examinations, end of semester design problem/project and student exit surveys are used as metrics to check efficacy of pedagogy.

ABET: a,b,e,g,i,k


AU  - Aneet Dharmavaram Narendranath
CY  - Columbus, Ohio
DA  - 2017/06/24
PB  - ASEE Conferences
TI  - Board # 85 : Introducing the Galerkin Method of Weighted Residuals into an Undergraduate Elective Course in Finite Element Methods
UR  - https://peer.asee.org/27941
DO  - 10.18260/1-2--27941
ER  -