Problem 37 - Concepts and Applications of Finite Element Analysis

Question

As an alternative to Eqs. 6.8-2 through \(6.8-4\), one can eliminate (say) \(\xi_{3}\) from shape functions \(N_{i}\) by use of the constraint relation \(\xi_{1}+\xi_{2}+\xi_{3}=1\), then take the derivatives \(\partial N_{i} / \partial \xi_{1}\) and \(\partial N_{i} / \partial \xi_{2} .\) Verify that this procedure also yields Eq. 6.8-5.

Step-by-Step Solution

Answer

By substituting \(\xi_{3} = 1 - \xi_{1} - \xi_{2}\) into each shape function and subsequently taking the derivatives of these modified functions with respect to \(\xi_{1}\) and \(\xi_{2}\), one can confirm that this procedure yields Eq. 6.8-5 just as Eqs. 6.8-2 through 6.8-4 do.

Step 1: Use the Constraint Relation

Shape functions are usually denoted as \(N_i\) where \(i\) is the node number. Given the shape functions \(N_1, N_2, N_3\) and the constraint relation \(\xi_{1} + \xi_{2} + \xi_{3} = 1\), one can eliminate \(\xi_{3}\) by substituting \(\xi_{3} = 1 - \xi_{1} - \xi_{2}\) into the shape functions.

Step 2: Form the Modified Shape Functions

Substitute the expression for \(\xi_{3}\) from the previous step into each shape function, which will yield the new modified shape functions that are now functions of \(\xi_{1}\) and \(\xi_{2}\) only.

Step 3: Take Derivatives

Take the derivative of each modified shape function with respect to \(\xi_{1}\) to get \(\partial N_{i} / \partial \xi_{1}\), then do similarly to get \(\partial N_{i} / \partial \xi_{2}\). These are the first order derivatives of our modified shape functions.

Step 4: Verify the Equation

Evaluate if the above derived expressions for the derivatives are equivalent to Eq. 6.8-5. If they match, it verifies that the alternative procedure to eliminate \(\xi_{3}\) also yields Eq. 6.8-5

Key Concepts

Shape Functions

Shape functions are essential in finite element analysis (FEM). They represent how unknown quantities like displacements, temperatures, or other field variables change within an element. In most applications, they are represented as functions of local coordinates. These functions are used to interpolate these quantities across the element using known values at nodes.

In the exercise, we discuss eliminating \(\xi_3\) from the set of shape functions \(N_i\). The constraint relation \(\xi_1 + \xi_2 + \xi_3 = 1\) helps reformulate the shape functions by expressing one variable in terms of others. By substituting \(\xi_3 = 1 - \xi_1 - \xi_2\), the shape functions depend only on \(\xi_1\) and \(\xi_2\). This substitution simplifies the computational process by reducing the dimensions in equations, making it easier to handle computational problems efficiently.

These modified shape functions allow us to work in two-dimensional space effectively. They form the foundation for analyzing complex systems by breaking them down into more manageable elements.

Numerical Methods

Numerical methods are techniques used to find approximate solutions to complex problems in mathematics and engineering. In the context of finite element analysis, numerical methods are crucial because they offer ways to analyze and simulate physical phenomena that are too complicated for analytical solutions.

In FEM, numerical methods assist in converting differential equations that describe the physical phenomena into a set of algebraic equations. These are more manageable and can be solved using powerful computational tools. This is particularly useful for handling problems involving complex geometries, material properties, and boundary conditions.

The exercise involves algebraic manipulation and substitution that is a classic example of numerical techniques. Modifying shape functions via elimination of \(\xi_3\) and differentiating accordingly are part of the systematic numerical approach in FEM. This allows engineers and scientists to effectively solve complex engineering problems that were once considered intractable.

Derivatives in FEM

Taking derivatives is a significant aspect of finite element analysis. Derivatives are used to understand how variables change in relation to one another, which is particularly important in physics and engineering problems.

In the context of the exercise, we take the derivative of the modified shape functions with respect to \(\xi_1\) and \(\xi_2\). The derivatives \(\frac{\partial N_i}{\partial \xi_1}\) and \(\frac{\partial N_i}{\partial \xi_2}\) help in formulating stiffness matrices and force vectors for finite element models. These derivative calculations are fundamental to determining how the interpolated field variables influence the overall behavior of the system.

Calculating derivatives accurately is crucial because it affects the precision of the finite element solution. This step ensures that the numerical solutions remain stable and dependable, forming a robust foundation for further simulation and analysis.