13.110. INFIN110 - 2D Infinite Solid

Matrix or VectorMapping and Shape Functions Integration Points
Magnetic Potential Coefficient Matrix Equation 11–144, Equation 11–147, and Equation 11–149 2 x 2
Thermal Conductivity and Specific Heat Matrices Equation 11–145, Equation 11–147, and Equation 11–149 2 x 2
Dielectric Permittivity and Electrical Conductivity Coefficient Matrices Equation 11–145, Equation 11–147, and Equation 11–149 2 x 2

References: Zienkiewicz et al.([170]), Damjanic' and Owen([171]), Marques and Owen([172]), Li et al.([174])

 

13.110.1. Mapping Functions

The theory for the infinite mapping functions is briefly summarized here. Consider the 1D situation shown below:

Figure 13.15: Global to Local Mapping of a 1D Infinite Element

Global to Local Mapping of a 1D Infinite Element

The 1D element may be thought of as one edge of the infinite element of Figure 13.16: Mapping of 2D Solid Infinite Element. It extends from node J, through node K to the point M at infinity and is mapped onto the parent element defined by the local coordinate system in the range -1 t 1.

Figure 13.16: Mapping of 2D Solid Infinite Element

Mapping of 2D Solid Infinite Element

The position of the "pole", xo, is arbitrary, and once chosen, the location of node K is defined by

(13–141)

The interpolation from local to global positions is performed as

(13–142)

where:

MJ(t) = -2t/(1 - t)
MK(t) = 1 - MJ(t)

Examining the above mapping, it can be seen that t = -1, 0, 1 correspond respectively to the global positions x = xJ, xK, , respectively.

The basic field variable is:

and can be interpolated using standard shape functions, which when written in polynomial form becomes

(13–143)

Solving Equation 13–142 for t yields

(13–144)

where:

r = distance from the pole, O, to a general point within the element
a = xK - xJ as shown in Figure 13.16: Mapping of 2D Solid Infinite Element

Substituting Equation 13–144 into Equation 13–143 gives

(13–145)

Where c0 = 0 is implied since the variable A is assumed to vanish at infinity.

Equation 13–145 is truncated at the quadratic (r2) term in the present implementation. Equation 13–145 also shows the role of the pole position, O.

In 2D (Figure 13.16: Mapping of 2D Solid Infinite Element) mapping is achieved by the shape function products. The mapping functions and the Lagrangian isoparametric shape functions for 2D and axisymmetric 4 node quadrilaterals are given in 2D and Axisymmetric 4-Node Quadrilateral Infinite Solids. The shape functions for the nodes M and N are not needed as the field variable, A, is assumed to vanish at infinity.

 

13.110.2. Matrices

The coefficient matrix can be written as:

(13–146)

with the terms defined below:

  1. Magnetic Vector Potential (accessed with KEYOPT(1) = 0)

    [Ke] = magnetic potential coefficient matrix
    μo = magnetic permeability of free space (input on EMUNIT command)

    The infinite elements can be used in magnetodynamic analysis even though these elements do not compute mass matrices. This is because air has negligible conductivity.

  2. Electric Potential (Electric Charge) (accessed with KEYOPT(1) = 1)

    [Ke] = dielectric permittivity matrix
    εx, εy = dielectric permittivity (input as PERX and PERY on MP command)
    = effective electrical conductivity (defined by Equation 5–83)

  3. Temperature (accessed with KEYOPT(1) = 2)

    [Ke] = thermal conductivity matrix
    kx, ky = thermal conductivities in the x and y direction (input as KXX and KYY on MP command)
    Cc = ρ Cp
    ρ = density of the fluid (input as DENS on MP command)
    Cp = specific heat of the fluid (input as C on MP command)
    {N} = shape functions given in 2D and Axisymmetric 4-Node Quadrilateral Infinite Solids

  4. Electric Potential (Electric Current) (accessed with KEYOPT(1) = 3)

    [Ke] = electrical conductivity matrix
    = effective electrical conductivity (defined by Equation 5–83)
    εx, εy = dielectric permittivity (input as PERX and PERY on MP command)

Although it is assumed that the nodal DOFs are zero at infinity, it is possible to solve thermal problems in which the nodal temperatures tend to some constant value, To, rather than zero. In that case, the temperature differential, θ (= T - To), may be thought to be posed as the nodal DOF. The actual temperature can then be easily found from T = θ + To. For transient analysis, θ must be zero at infinity t > 0, where t is time. Neumann boundary condition is automatically satisfied at infinity.

The {Bi} vectors of the [B] matrix in Equation 13–146 contain the derivatives of Ni with respect to the global coordinates which are evaluated according to

(13–147)

where:

[J] = Jacobian matrix which defines the geometric mapping

[J] is given by

(13–148)

The mapping functions [M] in terms of s and t are given in 2D and Axisymmetric 4-Node Quadrilateral Infinite Solids. The domain differential d(vol) must also be written in terms of the local coordinates, so that

(13–149)

Subject to the evaluation of {Bi} and d(vol), which involves the mapping functions, the element matrices [Ke] and [Ce] may now be computed in the standard manner using Gaussian quadrature.