Method of manufactured solutions
• The first stage in code verification is to test for problems for which we
have analytical solutions.
• However, often we have codes that are intended for more complex
problems than we can solve analytically.
• Codes typically solve differential equations for applied loadings in the
form of right hand sides and boundary conditions.
• If we cannot solve the differential equation for some representative
loading and boundary condition, we can specify a solution, calculate
the corresponding loading and boundary conditions, and use them as
input to the code.
Beam example
• The general equations of a beam loaded by a transverse load q(x)
d2
dx 2
 d 2w 
 EI dx 2   q ( x) 0  x  L


• We want to verify a code that solves this problem for any distribution
of EI.
• We can start by comparing the solution for constant EI, since for this
case it is easy to obtain an analytical solution.
• For example, for constant EI and constant q, integrating to obtain a
quartic polynomial is very easy, and solution can be found in
Mechanics of Materials textbooks.
Manufactured solution for beam
• We next want to test code for varying EI, for example EI=cos(x/L) with
clamped boundary conditions
d2
dx 2

d 2w 
cos( x / L) dx 2   q( x)


0 xL
w(0) 
dw
(0)  0
dx
• Instead of specifying load and trying to find an analytical solution, we
will specify the solution w=x2 and calculate the load
d2
2
q ( x)  2  2 cos( x / L)    2 cos( x / L)
dx
L
• We will now apply this load and check whether our code will produce
w=x2
General procedure (V&CSC 6.3.1)
1. Establish the mathematical model in the form L(u)=0, where L() is
the differential operator and u is the dependent variable.
2. Choose the analytic form of the manufactured solution um.
3. Operate the mathematical model L onto the manufactured solution
um and obtain the analytic source term s=L(um).
4. Obtain the modified mathematical model by including the analytic
source term L(u)=s
5. (Mine) check whether the code produces the manufactured
solution under the action of s.
Top Hat question
• You wrote a code to solve the algebraic equation cosx+7tan(ln(x))=a
For a series of values of a.
• What value of a will you test the code on in the spirit of
manufactured solutions?
• Assume that the equation gives numerical problems to your favorite
root solver, and this is why you wrote your own code to solve it.
Remarks
• Manufactured solutions do not need to be physically realistic, but
care must be taken that they do not crash the code (e.g. square root
of negative temperature).
• When multiple terms appear in the equations, manufactured
solutions should not make one term dominant because this will not
verify the other terms.
• Trigonometric and exponential functions are good for manufactured
solutions because they are infinitely differentiable.
• Having a manufactured solution that varies slowly will allow
convergence with coarse meshes, hence inexpensive.
Example of MMS with order verification
• 6.3.4.1 Steady state heat conduction
• A discrete solution described in 6.3.4.1
was proven to have second order
convergence, which needed to be
verified for the code implementing it.
• Manufactured solution that was used
x 
 y 
  xy 
Tm ( x, y )  400  45cos 
  35sin 
  27.7 sin 

 15 
 20 
 50 
0 x5
0 y3
• What is needed to generate the
manufactured solution besides the
source term?
 2T  2T

 s ( x, y )
x 2 y 2
Grids
• Five grids were used 9x9, 17x17, 33x33, 65x65, and 129x129 nodes.
• Grids are non-uniform to cater to how fast solution is changing.
Grid convergence
• Observed order of accuracy