Comparison of various numerical methods

Question 1

Finite Element Method - Advantages

Answer 1

• Most popular for modelling due to ease of use
• Plenty of commercial packages available
• Any shape of domain can be modelled, possibility of including holes, gaps, etc.
• Construction and/ or excavation of soil layers (staged or wished in place)

Question 2

Finite Element Method - Disadvantages

Answer 2

• Expertise required
• Multi-variable functional calculus required
• Expensive time-wise
• Deciding the number of increments in a non-linear and/ or plastic analysis

Question 3

Finite Difference Method - Advantages

Answer 3

• Easy to use
• Easy to understand mathematically
• Short computational time

Question 4

Finite Difference Method - Disadvantages

Answer 4

Doesn’t work as well as the FEM for complicated geometry

Question 5

Boundary Element Method

Answer 5

Uses elements along the boundary of the model, rather than throughout the model. It is now emerging as a viable alternative to finite difference and finite element methods for solving a wide range of engineering problems.

Question 6

Boundary Element Method - Advantages

Answer 6

• Designing a mesh only requires the boundary to be divided, unlike FEM, where the whole domain has to be considered.
• Only requires the surface of the model to be meshed. Hence, design changes are very easy to accommodate.
• the number of simultaneous equations to be solved in the system is much less than the FEM, so much more efficient and quicker for certain a type of problems.
• much better than other methods for unbounded domains such as infinite ground.

Question 7

Boundary Element Method - Disadvantages

Answer 7

• Not as general as the FEM.
• Requires a thorough understanding of the mathematical basis behind the method.
• There are not a lot of commercial programs around; they are mostly research-based. One good commercial program is GPBEST.
• Modelling multi-layered soils, problems such as those with body forces, time-dependent effects, or non-linearities become complex, and the overall efficiency of the method reduces drastically.
• Simultaneous equations are not symmetric or banded as when using the FEM for elastoplastic associated flow and elastic materials - which makes the method less efficient.
• Requires “fundamental” closed-form equations, which require rigorous mathematics to obtain. (Exact solutions)

Question 8

Discrete Element Method (DEM)

Answer 8

It treats the domain as a dis-continuum, where cracks, fractures, etc. may occur, and was developed for particulate matter materials such as solids or powders. The system is modelled as a large assembly of distinct interacting general-shaped (deformable or rigid) bodies or particles, such as quadrilaterals in 2D, circular discs in 3D, etc., that are subject to motion. The interaction between particles is linked together by determining the contact forces and the non-linear interaction between the bodies by solving the equations of motion.

Question 9

Discrete Element Method - Advantages

Answer 9

• Good for modelling deformations and failure mechanisms (cracks, fractures, e.g. hydraulic fracturing)
• Good for modelling highly dynamic effects such as fracturing
• Mainly used for rock mechanics and tunnelling - only recently used in soil mechanics

Question 10

Coupled FEM/ BEM

Answer 10

Regions of interest in a problem are modelled using the FEM, while BEM is used for other regions, where only the boundaries of these areas need to be discretised. Consistent infinitesimal finite-element cell method.

Question 11

Coupled FEM/ BEM - Advantages

Answer 11

• Combines the benefits of both FEM and BEM
• Doesn’t require mathematical “fundamental solutions” as in the BEM

Question 12

Coupled FEM/ BEM - Disadvantages

Answer 12

• Requires a solid mathematical background in order to apply the method to different applications
• Very little research has been done on it