HW Problem 1:
Summary of Martin J. Gander and Gerhard Wanner’s paper “From Euler, Ritz, and Galerkin
to Modern Computing*” section 3 and 4:
In engineering there are many problems modeled in partial differential equations, however
many of these differential equations cannot be solved in closed form
solution. Approximations are generally needed to solve these problems and are referred to
as the weak or variational form of the solution. This requires introducing arbitrary test or
approximate functions into the differential equation and integrating by parts. Good choice in
arbitrary functions will result in more accurate solutions. In general the more terms
introduced in the sum the more accurate the results will be. The variational form can be
then taken by introducing boundary conditions such as the dirichlet condition.
The deformation of an elastic plate was a very difficult problem to solve and was first
considered by Sophie Germaine and through the years corrections and improvements were
made. However it wasn’t until 1850 that a breakthrough was achieved by
Kirchhoff. Kirchhoff formed the double partial differential equation of the form below.
To obtain a unique solution to the above problem suitable boundary conditions are
necessary. These boundary conditions are listed below:
where ῼ denotes the domain range.
Ritz proposed solving the elastic plate problem by converting the above differential equation
into variational form. This was accomplished by using the “wie man ohne weiteres einsieht”
procedure. Once done, the following formulation is obtained:
The next step is to choose a sequence of functions to approximate the solution to the above
formulation as a linear combination.
Coefficients a1,a2..,am will need to be determined. The quality of the solution will depend
on the choice of ψi(x, y). ψi(x, y) should generally take into account the boundary
conditions of the problem being analyzed.
After the ψi(x, y) is obtained it can be reinserted into the J formulation. This will result in a
finite -dimensional expression in a1,a2..am.
Next J will be differentiated to obtain when J is minimum. This will result in a linear system
which may be tedious to solve by hand but easy with today’s computers. The coefficients
a1,a2..am are obtained as a result of this linear system. It should be noted that as m
approaches infinity the error of a will be reduced.
HW Problem 2:
Galerkin Finite Element Method is a technique to solve approximate solutions to these
problems. The steps to performing the Galerkin Finite Element Method are as follows:
1. Variational Weak Formulation
2. Finite Mesh and Basis Functions
3. Galerkin Method
4. Solve system of algebraic equations
5. quality of approximate solution
Engineering problems are generally represented in differential form. To use the Galerkin
Finite Element Method the differential equation will need to be written in its variational weak
form. To obtain a test function v(x) will be needed. v(x) shall lie inside realm of [0,1] such
that the second derivative of function u(x) with respect to x will equal a known function f. f is
generally from given data. To solve the differential equation for the unique problem a set of
boundary conditions will be necessary. These boundary conditions are generally in the
form of Neuman or Dirichlet B.C.s. Neumann B.C.s are those that have the differential of
u(x) =0. While Dirichlet will have some given value of x in u(x) equal to zero.
After introducing the test function v(x) it will be necessary multiply both RHS and LHS of the
differential equation by v(x). Generally the double differential will be written as a different
letter or different notation such that it is now a single differential. This simplification will
allow for the process of integration by parts (LHS only). After the simplification the BCs will
be introduced and the equation will be simplified. It should be noted that BCs of Dirichlet
type will require v(x) to be zero. The result is the weak formulation.
The next step is the mesh step where the domain will be subdivided into continuous
elements. Where elements meet will be called nodes.
The next sub-step is introducing the basis functions, generally in the form of Lagrange
Polynomials. These are interpolating polynomials to obtain basis or weight functions. Each
element will contain local finite element basis functions that will be used for the global basis
functions of the nodes. These functions are in the form of roof functions.
The third step is to introduce the Galerkin Method. While u(x) is desired, an approximate
solution uh(x) will be obtained. The test function v(x), will be represented as the summation
of the weighted nodal functions and uh(x) will be the summation of the product of ui and
vi(x).
These equations form a collection of terms in a system of equations that can be solved with
linear algebra. The system of equations is of the form KU=F, where K is the stiffness, U the
desired solution, and F the forcing function.
Once the desired solution is obtained the validity of the solution can be obtained by
comparing the approximate solution with known data. This will give the error.
HW Problem 3.
𝑑2 𝑢
= −𝑢′′ = 𝑓(𝑥)
𝑑𝑥 2
𝑢′ (0) = ℎ(𝑥)
𝑢(1) = 𝑔(𝑥)
𝑣(1) = 0 (𝑑𝑖𝑟𝑒𝑐ℎ𝑙𝑒𝑡)
1
1
∫ 𝑢′ 𝑣 ′ 𝑑𝑥 − {𝑣(1)𝑢′ (1) − 𝑣(0)𝑢′ (0)} = ∫ 𝑓(𝑥)𝑣(𝑥)𝑑𝑥
0
0
1
1
∫0 𝑢′ 𝑣 ′ 𝑑𝑥 − {𝑣(1)𝑢′ (1) − 𝑣(0)𝑢′ (0)} = ∫0 𝑓(𝑥)𝑣(𝑥)𝑑𝑥 : direchlet condition
1
1
∫ 𝑢′ 𝑣 ′ 𝑑𝑥 − {𝑣(0)ℎ(𝑥)} = ∫ 𝑓(𝑥)𝑣(𝑥)𝑑𝑥
0
0
HW Problem 4.
𝛻 2 𝑢 = 𝑓(𝑥, 𝑦)
𝑢=0
𝛻2𝑢 =
𝜕2𝑢 𝜕2𝑢
+
𝜕𝑥 2 𝜕𝑦 2
𝑏
𝑏
𝑏
𝑏
𝑏
𝑏
𝑏
𝑏
𝑏
𝑏
∫𝑎 ∫𝑎 ∇𝑢∇𝑣𝑑𝑥𝑑𝑦- {𝑣(1)𝑢′ (1) − 𝑣(0)𝑢′ (0)}=∫𝑎 ∫𝑎 𝑓(𝑥, 𝑦)𝑣(𝑥, 𝑦)𝑑𝑥𝑑𝑦
∫𝑎 ∫𝑎 ∇𝑢∇𝑣𝑑𝑥𝑑𝑦- {𝑣(1)𝑢′ (1) − 𝑣(0)𝑢′ (0)}=∫𝑎 ∫𝑎 𝑓(𝑥, 𝑦)𝑣(𝑥, 𝑦)𝑑𝑥𝑑𝑦 : direchlet condition
𝑏
𝑏
∫𝑎 ∫𝑎 ∇𝑢∇𝑣𝑑𝑥𝑑𝑦= ∫𝑎 ∫𝑎 𝑓(𝑥, 𝑦)𝑣(𝑥, 𝑦)𝑑𝑥𝑑𝑦
b) Energy functional
𝑏 𝑏
1
𝐽 = ∫ ∫ ( ( ∇𝑢)2 − 𝑓 ∗ 𝑢)dxdy − −>> minimum
𝑎 𝑎 2