Thermal Deformations of a 3-D Brick

Let us consider a block. A length of the block is L, a width is b, a height is h.
Sought quantities are the absolute elongations of the block along axes X,Y,Z because of a temperature change.
Let us use the following initial data: L = 0.3 m, b = 0.2 m, h = 0.1 m.
Material characteristics: the Young's modulus E = 2.1E+011 Pa, Poisson's ratio ν = 0.28, linear expansion coefficient α = 1.3E-005 K -1.

The temperature change ΔT is 100o .

Thermal Deformations of a 3-D Brick, the finite element model with applied loads and restraints

The finite element model with applied loads and restraints

The analytical solutions are calculated by the formulas:
Δx = α L ΔT
Δy = α b ΔT
Δz = α h ΔT

Thus,
Δx = 3.90000000E-004 m
Δy = 2.60000000E-004 m
Δz = 1.30000000E-004 m .

After carrying out calculation with the help of AutoFEM, the following results are obtained:

Table 1.Parameters of the finite element mesh

Finite Element Type	Number of Nodes	Number of Finite Elements
quadratic tetrahedron	449	1729

Table 2. Result "Displacement"

Numerical Solution Displacement Δ*, m	Analytical Solution Displacement Δ, m	Error δ =100%\|Δ - Δ \|/\| Δ \|
3.90000147E-004	3.90000000E-004	0.38E-004
2.60000059E-004	2.60000000E-004	0.23E-004
1.30000117E-004	1.30000000E-004	0.90E-004

Thermal Deformations of a 3-D Brick, Result "Displacement" of finite element analysis

Conclusions:

The relative error of the numerical solution compared to the analytical solution for displacements is equal to 0.0001% for quadratic finite elements.

*The results of numerical tests depend on the finite element mesh and may differ slightly from those given in the table.