Problem 40 - Concepts and Applications of Finite Element Analysis

Question

Many methods of solving large eigenproblems require factoring either the stiffness matrix or a combination of the stiffness and mass matrices (e.g.) the determinant search and subspace iteration methods). Factoring requires approximately \(n_{\mathrm{eq}} b^{2} / 2\) operations (i.e., multiplications) where \(n_{\mathrm{eq}}\) is the number of equations and \(b\) is the semibandwidth. For full matrices, the number of operations is about \(n_{e q}^{3} / 6 .\) Consider a system of 5000 equations with \(b=500\). If this system of equations is partitioned into \(m\) master and \(s\) slave d.o.f., what must \(m\) be so that factoring the condensed (full) system is no more expensive than factoring the original (banded) system? What if \(b=100\) instead?

Step-by-Step Solution

Answer

The value of m that makes factoring the condensed (full) system and the original (banded) system equally expensive in terms of operations is given by \(m = \sqrt[3]{\frac{n_{\mathrm{eq}} b^{2}}{3}}\). For the given numbers of \(n_{\mathrm{eq}}=5000\) and \(b=500\), the value of \(m\) will be approximately 2083. Similarly, when \(b=100\), the value of \(m\) will be roughly 185.

Step 1: Understand the problem and define the equality

The first step is to formulate a relationship for the number of operations needed for both types of matrices to be equal. For banded matrix factoring, the number of operations is given by \(\frac{n_{\mathrm{eq}} b^{2}}{2}\) and for full matrix factoring, the number of operations is \(\frac{n_{\mathrm{eq}}^{3}}{6}\). We aim to find \(m\), the number of master degrees of freedom (d.o.f), wherein \(n_{\mathrm{eq}} = m + s\), with \(s\) denoting the number of slave d.o.f. It is assumed that after partitioning, only the master d.o.f are factored, which results in a full matrix. Therefore, the equality becomes \(\frac{n_{\mathrm{eq}} b^{2}}{2} = \frac{m^{3}}{6}\)

Step 2: Rearrange the equation to solve for \(m\)

Starting from the equation defined in step 1, we can isolate \(m\) and solve for it. The rearranged equation is \(m = \sqrt[3]{\frac{n_{\mathrm{eq}} b^{2}}{3}}\)

Step 3: Substitute known values

We can now substitute the known values in each of the two given cases to solve for \(m\). The given values are \(n_{\mathrm{eq}}=5000\), \(b=500\) for the first case, and \(b=100\) for the second case.

Step 4: Calculate the value of \(m\) for each situation

Now, we put the values into the equation from Step 2, to solve for \(m\). For \(b=500\), \(m = \sqrt[3]{\frac{5000 \times 500^2}{3}}\) and for \(b=100\), \(m = \sqrt[3]{\frac{5000 \times 100^2}{3}}\). These solutions give the values of \(m\) which makes the factoring of the condensed (full) system and the original (banded) system equally expensive in terms of operations.

Key Concepts

Stiffness Matrix

The stiffness matrix is a fundamental concept in finite element analysis (FEA), representing how a structure resists deformation in response to applied forces. Think of it as a matrix that tells you how 'stiff' various parts of your structure really are.

In technical terms, for a given system with multiple degrees of freedom, the stiffness matrix is a square matrix where each entry represents the force along one degree of freedom due to a unit displacement along another degree of freedom, while all other degrees of freedom are held constant. It's critical for understanding how various elements of a structure will behave under stress, which is why we calculate it in FEA.

The concept can be easier to grasp by imagining a spring: The stiffer the spring, the more force is required to stretch it. Each spring has its own stiffness value, and together, these can make up the elements of a stiffness matrix in a larger system, mirroring the interconnected springs’ reactions to forces.

Eigenproblems

Eigenproblems are at the heart of many complex analyses in applied mathematics and engineering, particularly in vibrations, stability, and buckling analyses. In essence, an eigenproblem is a type of problem that asks for certain values (called eigenvalues) which result in non-trivial solutions from a set of linear equations, usually represented in matrix form.

When we solve eigenproblems in finite element analysis, we often look for natural frequencies and modes of structures. The stiffness and mass matrices play a central role here. Each eigenvalue corresponds to a natural frequency, and the associated eigenvector describes the mode shape at that frequency.

In more tangible terms, think about a guitar string that vibrates—each vibration mode is an eigenvector, and the frequency it vibrates at is the eigenvalue. The process of solving eigenproblems usually requires significant computational resources, and optimizing this process, like factoring matrices efficiently, is crucial in FEA.

Degrees of Freedom (d.o.f)

Degrees of freedom in the context of finite element analysis are the number of independent ways in which a dynamic system can move without violating any constraint imposed on it. Essentially, it's a measure of how 'free' a system is to change its state.

In structural engineering, for example, a point in space has three degrees of freedom for translation along the x, y, and z axes, and three degrees for rotation about these axes, giving it a total of six degrees of freedom.

When we partition a system into master and slave degrees of freedom, as mentioned in the example exercise, we're basically deciding which parts of our system we need to thoroughly analyze ('master') and which parts we can treat with simpler computations ('slave'). This partitioning is crucial because it can significantly reduce the complexity of a problem, making it more solvable—much like simplifying a complex, interwoven puzzle by organizing it into smaller, more manageable pieces.