Finite Element Method - 3D Mesh Generator - Metfem3D

Linear Examples

Diffusion equation

Putting k=0 in the equation of electrodiffusion we neglect the electrostatic term. It leads to the following equation describing diffusion in ℜⁿ ^[1,^2]

u_t - D Δu = 0 in ℜⁿ ✗ (0, ∞),
u = g on ℜⁿ ✗ t = 0.

where D denotes a diffusion coefficient. This kind of equation represents an initial value problem. Assuming that the considered domain Ω in ℜ³ is of a cubic type [x_min, x_max]✗[y_min, y_max]✗[z_min, z_max] let us take u = 0 as a boundary condition. Now we seek a solution of the equation which satisfies this boundary condition and prescribed initial condition at the time t = 0. The solution of the equation is approximated by the triple sum

	∞	∞	∞
u(x,y,z,t) =	∑	∑	∑	v_{0, k_x, k_y, k_z} e^{-(k_x² + k_y² + k_z²)Dt} sin(k_xx) sin(k_yy) sin(k_zz),
	k_x = 1	k_y = 1	k_z = 1

where v_{0, k_x, k_y, k_z} are unknown coefficients that must be determinated from the initial condition:

	∞	∞	∞
g = u(., 0) =	∑	∑	∑	v_{0, k_x, k_y, k_z} sin(k_xx) sin(k_yy) sin(k_zz).
	k_x = 1	k_y = 1	k_z = 1

In the case of the domain being [0 π] ✗ [0 π] ✗ [0 π] and g = const the solution has the form

	∞
u(x,y,z,t) = 64g/(π³)	∑	e^{-(k_x² + k_y² + k_z²)Dt} sin(k_xx) sin(k_yy) sin(k_zz)/(k_x k_y k_z).
	k_x, k_y, k_z = 1, 3, …

In the case of cylindrical domain defined by r ∈ [0, r₀], θ ∈ [0, 2π) and z ∈ [0, π], and with the boundary condition of the form u = 0 the solution of diffusion equation can be expanded in an absolutely and uniformly convergent series of the form

	∞	∞
u(r, θ, z, t) =	∑	∑	J_n(k_{n, m}r/r₀)	(	a_{n, m}cos(n θ) + b_{n, m} sin(nθ)	)	sin(k_z z)✗
	n = 0	m, k_z = 1

✗	e^{-((k_n,m/r₀)² + k_z²)Dt}

where k_{n, m} are the zeros of the Bessel functions and a_{n, m}, b_{n, m} are constants that must be found from the initial condition by making use of the orthogonality relation for the trigonometric and Bessel functions.

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Finite Element Method - 3D Mesh Generator - Metfem3D

Linear Examples

Diffusion equation

References