Page 12 - Finite Element Analysis with ANSYS Workbench

1.1 Solving Engineering Problems 3

(a) Differential Equations. The differential equations
interpret and model physical behavior of the problem into
mathematical functions. For example, if we would like to
determine temperature distribution of a ceramic cup containing hot
coffee, we need to solve the differential equation that describes the
conservation of energy at any location on the cup. The differential
equation contains partial derivative terms representing conduction
heat transfer inside the cup material. Such differential equation is
not easy to solve using analytical approaches.

(b) Boundary Conditions. The temperature distribution
on the cup depends on the coffee temperature inside the cup and the
surrounding ambient temperature outside the cup surface.
Different boundary conditions thus affect the cup temperature
solution.

(c) Geometry. Cup shapes also affect their temperature
distribution, even though they are made from the same material and
placed under the same boundary conditions. The cup temperature
changes if the cup is larger or thicker.
The three components above always affect the solutions
of the problem being solved. In undergraduate classes, we learned
how to solve simplified forms of differential equations subjected to
simple boundary conditions on plain geometries to obtain exact or
analytical solutions. For real-life practical problems, they are
governed by coupled differential equations which are quite
sophisticated. Their boundary conditions and geometries are
complicated. Numerical methods such as the finite element and
finite volume methods are employed to provide accurate
approximated solutions.

1.1.2 Solution Methods

Methods for finding solutions can be categorized into
two types:
(a) Analytical Method. The analytical method herein
refers to a mathematical technique used to find an exact or
analytical solution for a given problem. The technique can provide