Problem 19 - Concepts and Applications of Finite Element Analysis

Question

The differential equation for wind-driven circulation in a shallow lake is $$ \psi_{, x x}+\psi_{1 y y}+A \psi_{x}+B \psi_{y}+C=0 $$ where \(\psi\) is the stream function and \(A, B\), and \(C\) are functions of \(x\) and \(y\). With \(h=\) depth, depthwise average velocities are \(u=\psi, y h\) and \(v=\) \(-\psi, \gamma / h\). Coordinates \(x\) and \(y\) are tangent to the lake surface. The nonessential boundary condition is \(\psi_{y n}=0\) on the shoreline, where \(n\) is a direction normal to the shoreline. Derive a finite element formulation by the Galerkin method [15.8]. A symbolic result is desired, analogous to Eq. 15.6-7, not details of a particular element.

Step-by-Step Solution

Answer

Using the Galerkin method, the finite element formulation of this differential equation has been derived to be: \(\int_\Omega \boldsymbol{\nabla} \psi^h · \boldsymbol{\nabla}w d\Omega = \int_\Omega (A\psi_x + B\psi_y + C)w d\Omega\) where \(\Omega\) is the problem domain.

Step 1: Reformulate the Differential Equation

First, rewrite the given differential equation \(\psi_{xx} + \psi_{yy} + A\psi_x + B\psi_y + C = 0\) by moving terms with coefficients to the other side to get: \(-\psi_{xx} - \psi_{yy} = A\psi_x + B\psi_y + C\).

Step 2: Applying Galerkin Method

In the Galerkin finite element method, we approximate our solution \(\psi\) by a finite element solution. Let \(\psi^h\) be this finite element solution, defined by \(\psi^h = \sum_{i=1}^n a_i \phi_i\), with \(a_i\) coefficients and \(\phi_i\) basis functions. \n\nThen, apply the Galerkin method, and multiply the equation by a test function \(w\), then integrate over the problem domain, denoted by \(\Omega\), to obtain: \(\int_\Omega ( -\psi_{xx} - \psi_{yy})w d\Omega = \int_\Omega (A\psi_x + B\psi_y + C)w d\Omega\).

Step 3: Dealing with the Derivatives

To get rid of the second derivatives, apply the Green's formula, which states \(\int_\Omega ( -\psi_{xx} - \psi_{yy})w d\Omega = \int_\Omega \boldsymbol{\nabla} \psi^h · \boldsymbol{\nabla}w d\Omega + \oint_{\partial \Omega} \frac{\partial w}{\partial n} ds\), with \(∂Ω\) being the boundary of the lake and \(ds\) the differential length on the boundary. In this context, \(\frac{\partial w}{\partial n}\) represents the boundary condition.

Step 4: Applying Boundary Conditions

The given nonessential boundary condition states that \(\psi_{yn}=0\) at the shoreline. In our formulation, this boundary condition means that \(\frac{\partial w}{\partial n}\) equals zero. Considering this, the second term in the Green's formula becomes zero, leading to: \(\int_\Omega \boldsymbol{\nabla} \psi^h · \boldsymbol{\nabla}w d\Omega = \int_\Omega (A\psi_x + B\psi_y + C)w d\Omega\).

Key Concepts

Understanding the Galerkin Method

The Galerkin method is a powerful mathematical approach used in finite element analysis to solve differential equations, particularly those that arise in engineering and physics. The method is named after the Russian mathematician Boris Grigoryevich Galerkin.

It utilizes weighted residuals to approximate solutions by minimizing the error across a domain. To do so, it represents the unknown function (in this case, the stream function \(\psi\)) as a sum of weighted 'trial' functions or basis functions \(\phi_i\), each multiplied by a coefficient \(a_i\). These basis functions are chosen to satisfy the boundary conditions imposed on the problem.

After this, you would multiply the governing differential equation by a 'test' function, also known as a weighting function \(w\), and integrate over the entire domain \(\Omega\). This helps to ensure that the residual is orthogonal to the subspace spanned by the test functions, striving for a best 'fit' between the trial function and the true solution across the domain.

The Galerkin method is specifically useful because it accommodates complex geometries and boundary conditions, making it highly versatile for numerous applications, including the analysis of wind-driven circulation in lakes as presented in the exercise.

Demystifying Differential Equations

Differential equations are mathematical equations that relate some function with its derivatives. They play a central role in engineering, physics, and many other sciences as they are used to describe various phenomena such as heat conduction, wave motion, fluid dynamics, and, as seen in our exercise, wind-driven lake circulation.

In our context, the differential equation \(\psi_{xx} + \psi_{yy} + A\psi_x + B\psi_y + C = 0\) involves derivatives of a stream function \(\psi\) with respect to spatial coordinates \(x\) and \(y\). This equation represents the balance of forces at play in the system – the spatial changes (given by the second derivatives) and additional terms that may account for other factors like wind stress or water depth that are specific to the application.

Solving such an equation analytically is often challenging or impossible, especially for complex boundaries and variable coefficients \(A\), \(B\), and \(C\). That's where numerical methods like the Galerkin method step in, providing an approach to approximate the behavior of the lake's circulation without needing an explicit solution.

Applying Green's Formula

Green's formula, a crucial tool in calculus, relates the double integral over a domain \(\Omega\) to the line integral around the boundary \(\partial\Omega\). Specifically, it's used for converting integrals of divergence of gradient functions to surface integrals, which aids in addressing the second derivatives present in our differential equation.

In the problem at hand, Green's formula transforms the terms containing the second derivatives of the stream function \(\psi\). Instead of directly dealing with these second-order terms, we use Green's theorem to express them as first derivatives (gradient of \(\psi\)) and boundary integrals. However, because the boundary condition specifies that the derivative of \(\psi\) normal to the shore \(\psi_{yn} = 0\), the boundary integral vanishes, greatly simplifying the problem.

This step is pivotal because it reduces the complexity of the solution process, especially within the finite element framework where dealing with higher-order derivatives can become cumbersome. With Green's formula, the focus turns to the gradient functions, which are easier to approximate with the Galerkin method's finite element basis functions.