김경종
30 KiB
Radiation
The heat flow per unit area between corresponding points is assumed to be given by
q = F _ {A} (\theta_ {A} - \theta^ {Z}) ^ {4} - F _ {B} (\theta_ {B} - \theta^ {Z}) ^ {4},
where \theta ^ { Z } is the value of absolute zero temperature on the temperature scale being used; q is the heat flux per unit surface area crossing the gap at this point, from surface A to surface B ; \theta _ { A } and \theta _ { B } are the temperatures of the two surfaces; and F _ { A } and F _ { B } are constants.
The derivatives of q are
\frac {\partial q}{\theta_ {A}} = 4 F _ {A} (\theta_ {A} - \theta^ {Z}) ^ {3}
and
\frac {\partial q}{\theta_ {B}} = - 4 F _ {B} (\theta_ {B} - \theta^ {Z}) ^ {3}.
Jacobian matrix
The contribution to the variational statement of thermal equilibrium is
\delta \Pi = q A \delta \theta_ {A} - q A \delta \theta_ {B},
where A is the area. The contribution to the Jacobian matrix for the Newton solution is
\mathrm{d} \delta \Pi = \mathrm{d} q A \delta \theta_ {A} - \mathrm{d} q A \delta \theta_ {B},
where
\mathrm{d} q = \frac {\partial q}{\theta_ {A}} \mathrm{d} \theta_ {A} + \frac {\partial q}{\theta_ {B}} \mathrm{d} \theta_ {B}.
For "tied" thermal contact the temperature at point A is constrained to have the same temperature as point B. The Lagrange multiplier method is used to impose the constraint by augmenting the thermal equilibrium statement as follows:
\delta \Pi = \delta \lambda (\theta_ {A} - \theta_ {B}) + \lambda (\delta \theta_ {A} - \delta \theta_ {B}),
where ¸ is the Lagrange multiplier. The contribution to the Jacobian matrix for the Newton solution is
\mathrm{d} \delta \Pi = \delta \lambda (\mathrm{d} \theta_ {A} - \mathrm{d} \theta_ {B}) + \mathrm{d} \lambda (\delta \theta_ {A} - \delta \theta_ {B}).
5.2.5 Heat generation caused by frictional sliding
For coupled temperature-displacement analysis the *GAP HEAT GENERATION option in ABAQUS/Standard allows the introduction of a factor, \eta , which defines the conversion of frictional dissipation to heat. The fraction of generated heat into the first and second surface, f _ { 1 } and f _ { 2 } , respectively, can also be defined. This heat generation capability is to be used only in a coupled temperature-displacement analysis.
The heat fraction, ´, determines the fraction of the energy dissipated during frictional slip that enters the contacting bodies as heat. Heat is instantaneously conducted into each of the contacting bodies depending on the values of f _ { 1 } and f _ { 2 } . The contact interface is assumed to have no heat capacity and may have properties for the exchange of heat by conduction and radiation.
Refer to ``Small-sliding interaction between bodies, '' Section 5.1.1, and ``Finite-sliding interaction between deformable bodies,'' Section 5.1.2, for explanations of the notation used for the shape functions and contact parameters involved in the small-sliding and slide line theory. Note that the shape functions for interpolation of the temperature field may be different from the interpolation functions for the displacements; for example, if the underlying elements are of second order, the displacements are interpolated using quadratic functions, whereas the temperature field is interpolated using linear shape functions. Hence, the temperature interpolator will be denoted with the symbol M and the displacement interpolation will be denoted with the symbol N . Only the heat transfer terms will be discussed in the following.
The heat flux densities \mathbf { \tilde { \Pi } } ^ { -- } q _ { 1 } , going out the surface on side 1, and q _ { 2 } , going out the surface on side 2--are given by
q _ {1} = q _ {k} + q _ {r} - f _ {1} q _ {g}
and
q _ {2} = - q _ {k} - q _ {r} - f _ {2} q _ {g},
where q _ { g } is the heat flux density generated by the interface element due to frictional heat generation, q _ { k } is the heat flux due to conduction, and q _ { r } is the heat flux due to radiation.
The heat flux density generated by the interface element due to frictional heat generation is given by
q _ {g} = \eta \tau \dot {s} = \eta \tau \frac {\Delta s}{\Delta t},
where \tau is the frictional stress, \Delta s is the incremental slip, and \Delta t is the incremental time. The frictional stress is dependent on the contact pressure, p ; the friction coefficient, \mu ; and the temperatures on either side of the interface.
The heat flux due to conduction is assumed to be of the form
q _ {k} = \kappa (h, p, \overline {{\theta}}) (\theta_ {1} - \theta_ {2}) = \kappa (h, p, \overline {{\theta}}) \Delta \theta ,
Interface Modeling
where the heat transfer coefficient \kappa ( h , p , \overline { { \theta } } ) is a function of the average temperature at the contact point, \overline { { \theta } } = \textstyle { \frac { 1 } { 2 } } ( \theta _ { 1 } + \theta _ { 2 } ) ; the overclosure, h; and the contact pressure, p. \theta _ { 1 } and \theta _ { 2 } are the temperatures of side 1 and side 2, respectively.
The heat flux due to radiation is assumed to be of the form
q _ {r} = F _ {1} (\theta_ {1} - \theta^ {Z}) ^ {4} - F _ {2} (\theta_ {2} - \theta^ {Z}) ^ {4},
where F _ { 1 } and F _ { 2 } are the constants defined on the data line for the *GAP RADIATION option and \theta ^ { Z } is the absolute zero on the temperature scale used.
Using the Galerkin method the weak form of the equations can be written as
\int_ {S} \delta \theta_ {1} q _ {1} d S = \int_ {S} \delta \theta_ {1} (q _ {k} + q _ {r} - f _ {1} q _ {g}) d S, \qquad \int_ {S} \delta \theta_ {2} q _ {2} d S = \int_ {S} \delta \theta_ {2} (- q _ {k} - q _ {r} - f _ {2} q _ {g}) d S.
The contribution to the variational statement of thermal equilibrium is
\delta \Pi = \int_ {S} (\delta \theta_ {1} q _ {1} + \delta \theta_ {2} q _ {2}) d S = \int_ {S} [ \delta \Delta \theta (q _ {k} + q _ {r}) - \delta \hat {\theta} q _ {g} ] d S,
where \hat { \theta } = f _ { 1 } \theta _ { 1 } + f _ { 2 } \theta _ { 2 } . The contribution to the Jacobian matrix for the Newton solution is
Equation 5.2.5-1
d \delta \Pi = \int_ {S} [ d \delta \Delta \theta (q _ {k} + q _ {r}) + \delta \Delta \theta (d q _ {k} + d q _ {r}) - d \delta \hat {\theta} q _ {g} - \delta \hat {\theta} d q _ {g} ] d S.
At a contact point the temperatures can be interpolated with
\theta_ {1} (s) = M _ {1} ^ {N} (s) \theta^ {N}, \qquad \theta_ {2} (s) = M _ {2} ^ {N} (s) \theta^ {N},
where \theta ^ { N } is the temperature at the N th node associated with the interface element. Note that the summation convention will be used for all superscripts. Therefore, the temperature variables can be written as follows:
\Delta \theta (s) = \Delta M ^ {N} (s) \theta^ {N}, \qquad \hat {\theta} (s) = \hat {M} ^ {N} (s) \theta^ {N}, \qquad \overline {{\theta}} (s) = \overline {{M}} ^ {N} (s) \theta^ {N},
where \hat { M } ^ { N } ( s ) = f _ { 1 } M _ { 1 } ^ { N } ( s ) + f _ { 2 } M _ { 2 } ^ { N } ( s ) and \overline { { { M } } } ^ { N } ( s ) = \textstyle { \frac { 1 } { 2 } } ( M _ { 1 } ^ { N } ( s ) + M _ { 2 } ^ { N } ( s ) ) . Substituting the above expressions into Equation 5.2.5-1, we obtain
Interface Modeling
\begin{array}{l} d \delta \Pi = \delta \theta^ {N} \int_ {S} \left[ \left(q _ {k} + q _ {r}\right) \frac {d \Delta M ^ {N}}{d s} d s + \Delta M ^ {N} \left(\frac {\partial q _ {r}}{\partial \theta_ {1}} d \theta_ {1} + \frac {\partial q _ {r}}{\partial \theta_ {2}} d \theta_ {2} \right. \right. \\ + \frac {\partial q _ {k}}{\partial \Delta \theta} d \Delta \theta + \frac {\partial q _ {k}}{\partial \overline {{{\theta}}}} d \overline {{{\theta}}} + \frac {\partial q _ {k}}{\partial h} d h + \frac {\partial q _ {k}}{\partial p} d p \Big) \\ \left. - q _ {g} \frac {d \hat {M} ^ {N}}{d s} d s - \hat {M} ^ {N} \left(\frac {\partial q _ {g}}{\partial \tau} d \tau + \frac {\partial q _ {g}}{\partial s} d s\right) \right] d S. \\ \end{array}
After rearranging and expanding terms, we obtain
\begin{array}{l} d \delta \Pi = \delta \theta^ {N} \int_ {S} \left[ \left((q _ {k} + q _ {r}) \frac {d \Delta M ^ {N}}{d s} - q _ {g} \frac {d \hat {M} ^ {N}}{d s}\right) d s \right. \\ + \Delta M ^ {N} \left(\frac {\partial q _ {r}}{\partial \theta_ {1}} M _ {1} ^ {M} + \frac {\partial q _ {r}}{\partial \theta_ {2}} M _ {2} ^ {M} + \frac {\partial q _ {k}}{\partial \Delta \theta} \Delta M ^ {M} + \frac {\partial q _ {k}}{\partial \overline {{\theta}}} \overline {{M}} ^ {M}\right) d \theta^ {M} \\ + \Delta M ^ {N} \left(\frac {\partial q _ {k}}{\partial h} d h + \frac {\partial q _ {k}}{\partial p} d p\right) \\ + \Delta M ^ {N} \left(\frac {\partial q _ {r}}{\partial \theta_ {1}} \frac {d M _ {1} ^ {K}}{d s} \theta^ {K} + \frac {\partial q _ {r}}{\partial \theta_ {2}} \frac {d M _ {2} ^ {K}}{d s} \theta^ {K} + \frac {\partial q _ {k}}{\partial \Delta \theta} \frac {d \Delta M ^ {K}}{d s} \theta^ {K} + \frac {\partial q _ {k}}{\partial \overline {{\theta}}} \frac {d \overline {{M}} ^ {K}}{d s} \theta^ {K}\right) d s \\ - \left. \hat {M} ^ {N} \left(\frac {\partial q _ {g}}{\partial \tau} (\frac {\partial \tau}{\partial p} d p + \frac {\partial \tau}{\partial \theta_ {1}} d \theta_ {1} + \frac {\partial \tau}{\partial \theta_ {2}} d \theta_ {2}) + \frac {\partial q _ {g}}{\partial s} d s\right) \right] d S. \\ \end{array}
Expanding the terms involving frictional heat generation yields
\begin{array}{l} d \delta \Pi = \delta \theta^ {N} \int_ {S} \left[ \left((q _ {k} + q _ {r}) \frac {d \Delta M ^ {N}}{d s} - q _ {g} \frac {d \hat {M} ^ {N}}{d s}\right) d s \right. \\ + \Delta M ^ {N} \left(\frac {\partial q _ {r}}{\partial \theta_ {1}} M _ {1} ^ {M} + \frac {\partial q _ {r}}{\partial \theta_ {2}} M _ {2} ^ {M} + \frac {\partial q _ {k}}{\partial \Delta \theta} \Delta M ^ {M} + \frac {\partial q _ {k}}{\partial \overline {{\theta}}} \overline {{M}} ^ {M}\right) d \theta^ {M} \\ + \Delta M ^ {N} \left(\frac {\partial q _ {k}}{\partial h} d h + \frac {\partial q _ {k}}{\partial p} d p\right) \\ + \Delta M ^ {N} \left(\frac {\partial q _ {r}}{\partial \theta_ {1}} \frac {d M _ {1} ^ {K}}{d s} \theta^ {K} + \frac {\partial q _ {r}}{\partial \theta_ {2}} \frac {d M _ {2} ^ {K}}{d s} \theta^ {K} + \frac {\partial q _ {k}}{\partial \Delta \theta} \frac {d \Delta M ^ {K}}{d s} \theta^ {K} + \frac {\partial q _ {k}}{\partial \overline {{\theta}}} \frac {d \overline {{M}} ^ {K}}{d s} \theta^ {K}\right) d s \\ - \hat {M} ^ {N} \frac {\partial q _ {g}}{\partial \tau} \left(\frac {\partial \tau}{\partial \theta_ {1}} M _ {1} ^ {M} + \frac {\partial \tau}{\partial \theta_ {2}} M _ {2} ^ {M}\right) d \theta^ {M} \\ - \hat {M} ^ {N} \frac {\partial q _ {g}}{\partial \tau} \left(\frac {\partial \tau}{\partial \theta_ {1}} \frac {d M _ {1} ^ {K}}{d s} \theta^ {K} + \frac {\partial \tau}{\partial \theta_ {2}} \frac {d M _ {2} ^ {K}}{d s} \theta^ {K}\right) d s \\ - \left. \hat {M} ^ {N} \left(\frac {\partial q _ {g}}{\partial \tau} \frac {\partial \tau}{\partial p} d p + \frac {\partial q _ {g}}{\partial s} d s\right) \right] d S. \\ \end{array}
The derivatives of q _ { r } , q _ { k } , and q _ { g } , are as follows:
\frac {\partial q _ {r}}{\partial \theta_ {1}} = 4 F _ {1} (\theta_ {1} - \theta^ {Z}) ^ {3}, \quad \frac {\partial q _ {r}}{\partial \theta_ {2}} = - 4 F _ {2} (\theta_ {2} - \theta^ {Z}) ^ {3},
\frac {\partial q _ {k}}{\partial \Delta \theta} = \kappa (h, p, \overline {{\theta}}), \quad \frac {\partial q _ {k}}{\partial \overline {{\theta}}} = \frac {\partial \kappa (h , p , \overline {{\theta}})}{\partial \overline {{\theta}}} \Delta \theta ,
\frac {\partial q _ {k}}{\partial h} = \frac {\partial \kappa (h , p , \overline {{\theta}})}{\partial h} \Delta \theta , \quad \frac {\partial q _ {k}}{\partial p} = \frac {\partial \kappa (h , p , \overline {{\theta}})}{\partial p} \Delta \theta ,
\frac {\partial q _ {g}}{\partial \tau} \frac {\partial \tau}{\partial p} = \eta \frac {| \Delta s |}{\Delta t} \mu , \quad \frac {\partial q _ {g}}{\partial s} = \eta \frac {\tau}{\Delta t}.
For contact pairs and slide line elements, each integration point is associated with a unique slave node. If we associate M _ { 1 } ^ { N } with the slave surface, then M _ { 1 } ^ { N } will again have only a single nonzero entry equal to one and the derivatives of M _ { 1 } ^ { N } with respect to s vanish. In contrast, on the master surface there will be multiple nonzero entries in M _ { 2 } ^ { N } , which are a function of the position on the master surface at which contact occurs.
For GAPUNIT and DGAP elements each contact (integration point) is directly associated with a node pair. Hence, M _ { 1 } ^ { N } and M _ { 2 } ^ { N } each have one nonzero entry that is equal to one, and all terms involving derivatives of M _ { 1 } ^ { N } and M _ { 2 } ^ { N } with respect to s vanish.
The variations of overclosure, h, and slip, s, can be written as linear functions of the variations of displacement. These expressions, which determine the form of the \beta matrix for contact elements, have been derived in ``Small-sliding interaction between bodies, '' Section 5.1.1, and ``Finite-sliding interaction between deformable bodies,'' Section 5.1.2.
5.2.6 Heat generation caused by electrical current
For coupled electrical-thermal analysis in ABAQUS/Standard, the *GAP HEAT GENERATION option allows the introduction of a factor, ´, which defines the conversion of electrical dissipation to heat due to electrical current flowing across an interface. The option also allows definition of the fraction of generated heat into the first and second surface, f _ { 1 } and f _ { 2 } , respectively.
The heat fraction, ´, determines the fraction of the energy dissipated due to electrical current that enters the contacting bodies as heat. Heat is instantaneously conducted into each of the contacting bodies depending on the values of f _ { 1 } and f _ { 2 } . The contact interface is assumed to have no heat capacity and may have properties for the exchange of heat by conduction and radiation.
The heat flux densities, q _ { 1 } , going out the surface on side 1, and q _ { 2 } , going out the surface on side 2, are given by
q _ {1} = q _ {k} + q _ {r} - f _ {1} q _ {g}
and
Interface Modeling
q _ {2} = - q _ {k} - q _ {r} - f _ {2} q _ {g},
where q _ { g } is the heat flux density generated by the interface element due to electrical current, q _ { k } is the heat flux due to conduction, and q _ { r } is the heat flux due to radiation.
The heat flux due to conduction is assumed to be of the form
q _ {k} = \kappa (h, \overline {{\theta}}) (\theta_ {1} - \theta_ {2}) = \kappa (h, \overline {{\theta}}) \Delta \theta ,
where the heat transfer coefficient \kappa ( h , \overline { { \theta } } ) is a function of the average temperature at the contact point, \overline { { \theta } } = \textstyle { \frac { 1 } { 2 } } ( \theta _ { 1 } + \theta _ { 2 } ) , and overclosure, h. \theta _ { 1 } and \theta _ { 2 } are the temperatures of side 1 and side ^ { 2 , } respectively.
The heat flux due to radiation is assumed to be of the form
q _ {r} = F _ {1} (\theta_ {1} - \theta^ {Z}) ^ {4} - F _ {2} (\theta_ {2} - \theta^ {Z}) ^ {4},
where F _ { 1 } and F _ { 2 } are the constants defined on the data line below the *GAP RADIATION option, and \theta ^ { Z } is the absolute zero on the temperature scale used.
The electrical flux density, J , in the interface element is given in terms of the difference in the electric potential, \Delta \varphi , across the interface:
J = \sigma_ {g} (h, \overline {{\theta}}) (\varphi_ {1} - \varphi_ {2}) = \sigma_ {g} (h, \overline {{\theta}}) \Delta \varphi ,
where the gap electrical conductance \sigma _ { g } ( h , \overline { { \theta } } ) is a function of the overclosure, h , and the average temperature at the contact point, \overline { { \theta } } . \varphi _ { 1 } and \varphi _ { 2 } are the electric potentials of side 1 and side 2, respectively.
In a steady-state analysis the heat flux density generated by the interface element due to electrical current is given by
q _ {g} = \eta \sigma_ {g} (h, \overline {{\theta}}) (\varphi_ {1} - \varphi_ {2}) ^ {2} = \eta \sigma_ {g} (h, \overline {{\theta}}) (\Delta \varphi) ^ {2},
where \eta is the fraction of dissipated energy converted to heat. In a transient analysis the average heat flux density is given by
\begin{array}{l} q _ {g} = \frac {\eta}{\Delta t} \int_ {t} ^ {t + \Delta t} \sigma_ {g} (h, \overline {{\theta}}) (\Delta \varphi) ^ {2} d t \\ \approx \frac {1}{3} \eta \sigma_ {g} (h, \overline {{\theta}}) \left[ (\Delta \varphi_ {t + \Delta t}) ^ {2} + \Delta \varphi_ {t + \Delta t} \Delta \varphi_ {t} + (\Delta \varphi_ {t}) ^ {2} \right], \\ \end{array}
where t is the time at the start of an increment and \Delta t is the time increment.
Using the Galerkin method, the weak form of the equations can be written as
\int_ {S} \delta \theta_ {1} q _ {1} d S = \int_ {S} \delta \theta_ {1} (q _ {k} + q _ {r} - f _ {1} q _ {g}) d S, \qquad \int_ {S} \delta \theta_ {2} q _ {2} d S = \int_ {S} \delta \theta_ {2} (- q _ {k} - q _ {r} - f _ {2} q _ {g}) d S.
The contribution to the variational statement of thermal equilibrium is
\delta \Pi = \int_ {S} (\delta \theta_ {1} q _ {1} + \delta \theta_ {2} q _ {2}) d S = \int_ {S} [ \delta \Delta \theta (q _ {k} + q _ {r}) - \delta \hat {\theta} q _ {g} ] d S,
where \hat { \theta } = f _ { 1 } \theta _ { 1 } + f _ { 2 } \theta _ { 2 } . The contribution to the Jacobian matrix for the Newton solution is
Equation 5.2.6-1
d \delta \Pi = \int_ {S} [ \delta \Delta \theta (d q _ {k} + d q _ {r}) - \delta \hat {\theta} d q _ {g} ] d S.
At a contact point the temperatures can be interpolated with
\theta_ {1} (s) = M _ {1} ^ {N} (s) \theta^ {N} \qquad \mathrm{and} \qquad \theta_ {2} (s) = M _ {2} ^ {N} (s) \theta^ {N},
where \theta ^ { N } is the temperature at the N th node associated with the interface element. Note that the summation convention will be used for all superscripts. Therefore, the temperature variables can be written as follows:
\Delta \theta (s) = \Delta M ^ {N} (s) \theta^ {N}, \qquad \hat {\theta} (s) = \hat {M} ^ {N} (s) \theta^ {N}, \qquad \overline {{\theta}} (s) = \overline {{M}} ^ {N} (s) \theta^ {N},
where \hat { M } ^ { N } ( s ) = f _ { 1 } M _ { 1 } ^ { N } ( s ) + f _ { 2 } M _ { 2 } ^ { N } ( s ) and \overline { { { M } } } ^ { N } ( s ) = \textstyle { \frac { 1 } { 2 } } ( M _ { 1 } ^ { N } ( s ) + M _ { 2 } ^ { N } ( s ) ) . Substituting the above expressions to Equation 5.2.6-1, we obtain
\begin{array}{l} d \delta \Pi = \delta \theta^ {N} \int_ {S} \Big [ \Delta M ^ {N} \left(\frac {\partial q _ {r}}{\partial \theta_ {1}} d \theta_ {1} + \frac {\partial q _ {r}}{\partial \theta_ {2}} d \theta_ {2} + \frac {\partial q _ {k}}{\partial \Delta \theta} d \Delta \theta + \frac {\partial q _ {k}}{\partial \overline {{\theta}}} d \overline {{\theta}}\right) \\ - \left. \hat {M} ^ {N} \left(\frac {\partial q _ {g}}{\partial \Delta \varphi} d \Delta \varphi + \frac {\partial q _ {g}}{\partial \overline {{{\theta}}}} d \overline {{{\theta}}}\right) \right] d S. \\ \end{array}
The derivatives of q _ { r } , q _ { k } , and q _ { g } , are as follows:
\frac {\partial q _ {r}}{\partial \theta_ {1}} = 4 F _ {1} (\theta_ {1} - \theta^ {Z}) ^ {3}, \quad \frac {\partial q _ {r}}{\partial \theta_ {2}} = - 4 F _ {2} (\theta_ {2} - \theta^ {Z}) ^ {3},
\frac {\partial q _ {k}}{\partial \Delta \theta} = \kappa (h, p, \overline {{\theta}}), \quad \frac {\partial q _ {k}}{\partial \overline {{\theta}}} = \frac {\partial \kappa (h , p , \overline {{\theta}})}{\partial \overline {{\theta}}} \Delta \theta ,
and--in a steady-state analysis--
\frac {\partial q _ {g}}{\partial \Delta \varphi} = 2 \eta \sigma_ {g} (h, \overline {{\theta}}) \Delta \varphi , \qquad \frac {\partial q _ {g}}{\partial \overline {{\theta}}} = \eta \frac {\partial \sigma_ {g} (h , \overline {{\theta}})}{\partial \overline {{\theta}}} (\Delta \varphi) ^ {2},
while--in a transient analysis--
Interface Modeling
\frac {\partial q _ {g}}{\partial \Delta \varphi} = \frac {1}{3} \eta \sigma_ {g} (h, \overline {{\theta}}) (2 \Delta \varphi + \Delta \varphi_ {t}), \quad \frac {\partial q _ {g}}{\partial \overline {{\theta}}} = \frac {1}{3} \eta \frac {\partial \sigma_ {g} (h , \overline {{\theta}})}{\partial \overline {{\theta}}} \left[ (\Delta \varphi) ^ {2} + \Delta \varphi \Delta \varphi_ {t} + (\Delta \varphi_ {t}) ^ {2} \right].
For small-sliding interface elements, each integration point is associated with a unique slave node. If we associate M _ { 1 } ^ { N } with the slave surface, then M _ { 1 } ^ { N } will have only a single nonzero entry equal to one. In contrast, on the master surface, there will be multiple nonzero entries in M _ { 2 } ^ { N } .
Similarly, the contribution to the variational statement of electrical equilibrium is
\delta \Pi = \int_ {S} (\delta \varphi_ {1} J - \delta \varphi_ {2} J) d S = \int_ {S} \delta \Delta \varphi J d S,
and the contribution to the Jacobian matrix for the Newton solution is
d \delta \Pi = \int_ {S} \delta \Delta \varphi \left(\frac {\partial J}{\partial \overline {{\theta}}} d \overline {{\theta}} + \frac {\partial J}{\partial \Delta \varphi} d \Delta \varphi\right) d S.
The derivatives of J are
\frac {\partial J}{\partial \overline {{\theta}}} = \frac {\partial \sigma_ {g} (h , \overline {{\theta}})}{\partial \overline {{\theta}}} \Delta \varphi \qquad \mathrm{and} \qquad \frac {\partial J}{\partial \Delta \varphi} = \sigma_ {g} (h, \overline {{\theta}}).
5.2.7 Surface-based acoustic-structural medium interaction
ABAQUS/Standard provides two alternatives for modeling interaction between acoustic and structural media: a surface-based capability, which uses the *TIE option, or the use of acoustic interface elements (ASIn). If the special-purpose interface elements (ASIn) are used, interacting structural and acoustic nodes must be shared by the two meshes. The surface-based capability can be used for structural and acoustic meshes whose surface meshes are not spatially coincident or that have different node numbering. The ease of use and low computational cost of the surface-based procedure make it preferable to the element-based approach.
The equations on the contact interface between the structure and the acoustic medium
The method used in the *TIE option is a generalized way in which the tractions and volumetric acceleration fluxes are computed between structural and acoustic media. In place of consistent distributed tractions or fluxes on both media, one side (identified as the "slave") receives point tractions/fluxes based on interpolation with the shape functions from the other ("master") side. Either the acoustic fluid or the structural solid can be the slave or master, and no Lagrange multipliers are introduced in the formulation. The basis for deciding which to make slave or master is discussed in ``Acoustic and coupled acoustic-structural analysis, '' Section 6.9.1 of the ABAQUS/Standard User's Manual.
Interface Modeling
Transient expressions for the coupled acoustic-structural problem are:
\begin{array}{l} \int_ {V _ {f}} \left[ \delta p \left(\frac {1}{\mathrm{K} _ {f}} \ddot {p} + \frac {r}{\rho_ {f} \mathrm{K} _ {f}} \dot {p}\right) + \frac {1}{\rho_ {f}} \frac {\partial \delta p}{\partial \mathbf {x}} \cdot \frac {\partial p}{\partial \mathbf {x}} \right] d V + \int_ {S _ {\mathrm{fs}} \cup S _ {\mathrm{frs}}} \delta p \mathbf {n} ^ {-} \cdot \frac {\partial p}{\partial \mathbf {x}} d S - \int_ {S _ {\mathrm{ft}}} \delta p a _ {i n} d S \\ + \int_ {S _ {\mathrm{fr}} \cup S _ {\mathrm{frs}}} \delta p \left(\frac {r}{\rho_ {f}} \frac {1}{c _ {1}} p + \left(\frac {r}{\rho_ {f}} \frac {1}{k _ {1}} + \frac {1}{c _ {1}}\right) \dot {p} + \frac {1}{k _ {1}} \ddot {p}\right) d S = 0 \\ \end{array}
(acoustic medium) and
\begin{array}{l} \int_ {V} \delta \pmb {\varepsilon}: \pmb {\sigma} d V + \int_ {V} \alpha_ {c} \rho \delta \mathbf {u} ^ {m} \cdot \dot {\mathbf {u}} ^ {m} d V + \int_ {V} \rho \delta \mathbf {u} ^ {m} \cdot \ddot {\mathbf {u}} ^ {m} d V \\ + \int_ {S _ {\mathrm{fs}} \cup S _ {\mathrm{frs}}} \delta \mathbf {u} ^ {m} \cdot \mathbf {n} ^ {-} p d S - \int_ {S _ {\mathrm{t}}} \delta \mathbf {u} ^ {m} \cdot \mathbf {t} d S = 0 \\ \end{array}
(structural medium), where \mathbf { n } ^ { - } is the normal vector pointing into the fluid. The fluid-solid surface consists of the union of the directly coupled fluid-solid region, S _ { \mathrm { f s } } , and a region coupled via a "reactive" acoustic surface or impedance boundary, S _ { \mathrm { f r s } } . Of primary interest here are those terms integrated over S _ { \mathrm { f s } } \cup S _ { \mathrm { f r s } } , which couple the two variational equations. The fluid impedance integral, over S _ { \mathrm { f r } } \cup S _ { \mathrm { f r s } } , depends only on the acoustic pressure field and its variations, so it is unaffected by the contact with the solid. The derivation for the steady-state case is formally identical to the transient case and will not be discussed here. For details of the differences in transient and steady-state acoustics in ABAQUS, see ``Coupled acoustic-structural medium analysis,'' Section 2.9.1.
When ASIn elements are used (see ``Acoustic interface elements,'' Section 22.5.1 of the ABAQUS/Standard User's Manual), the formulation requires that the fluid and solid elements be geometrically and nodally conformal so that the shape functions for the structural displacements and the acoustic pressures are identical. The shape functions are integrated using standard methods to yield element matrices of dimension equal to the number of surface nodes on the element. The complete fluid-solid coupling matrices are formed by the sum over the element faces; that is, a standard element assembly operation. The two final coupling matrices have the sparsity pattern of the coupled fluid-solid element faces.
In surface-based coupling the interaction surface S _ { \mathrm { f s } } \cup S _ { \mathrm { f r s } } is formed by the boundary between possibly nonconforming structural and acoustic meshes. Therefore, the fluid-solid coupling matrix cannot be broken up into a sum over element faces as simply as in the ASIn case. To derive the coupling matrices in the surface-based procedure, we use a variation of the master-slave procedure used in small-sliding contact (see ``Small-sliding interaction between bodies, '' Section 5.1.1). At the start of an analysis, the projections \mathbf { x } _ { \mathrm { N } } of slave nodes onto the master surface are found, and the areas A _ { N } and normals n¡ \left( { \bf x } _ { \mathrm { N } } \right) associated with the slave nodes are computed. The projections are points { \bf p } \left( { \bf x } _ { \mathrm { N } } \right) on the master surface; master nodes in the vicinity of this projection are identified. Variables at the slave nodes \mathbf { x } _ { \mathrm { N } } are then interpolated from variables at the identified master surface nodes near the projection { \bf p } \left( { \bf x } _ { \mathrm { N } } \right) .
Since the physical degrees of freedom for the fluid and solid meshes are different, two cases must be treated. The two cases handle the discretization of the coupling terms differently.
Solid/structural master, fluid slave
If the fluid medium surface is designated as the slave, we constrain values at each fluid node to be an average of the values at nearby master surface nodes (see Figure 5.2.7-1). The pointwise fluid-solid coupling condition,
\frac {1}{\rho_ {f}} \frac {\partial p}{\partial \mathbf {x}} \cdot \mathbf {n} ^ {-} + \ddot {\mathbf {u}} ^ {m} \cdot \mathbf {n} ^ {-} = 0,
is enforced at the slave nodes, resulting in displacement degrees of freedom added to the fluid slave surface. These slave displacements are constrained by the master displacements and thereby eliminated; the slave pressures are not constrained directly.
Figure 5.2.7-1 Fluid slave.
text_image
n(x_N) x_{N-1} x_N u_2 A_N x_{N+1} u_1 u_2 p u_1 u_2 i+1 u_1 P(x_N) solid master surface fluid slave surface
Hence, the fluid equation coupling term
\int_ {S _ {\mathrm{fs}} \cup S _ {\mathrm{frs}}} \delta p \mathbf {n} ^ {-} \cdot \frac {\partial p}{\partial \mathbf {x}} d S
is equal to
- \int_ {S _ {\mathrm{fs}} \cup S _ {\mathrm{frs}}} \delta p \mathbf {n} ^ {-} \cdot \ddot {\mathbf {u}} ^ {m} d S.
This term is now approximated, at the slave node level, by the interpolated values of structural displacements at the nearby master nodes times the area of the slave node:
\int_ {S _ {\mathrm{fs}} \cup S _ {\mathrm{frs}}} \delta p \mathbf {n} ^ {-} \cdot \ddot {\mathbf {u}} ^ {m} d S \approx A _ {N} \left[ \sum_ {i} \mathbf {n} ^ {-} \left(\mathbf {x} _ {\mathrm{N}}\right) \cdot \mathrm{N} ^ {i} \left(\mathbf {p} \left(\mathbf {x} _ {\mathrm{N}}\right)\right) \ddot {\mathbf {u}} _ {i} ^ {m} \right],
where \mathrm { N } ^ { i } \left( \mathbf { p } \left( \mathbf { x } _ { \mathrm { N } } \right) \right) is the master surface interpolant evaluated at the projection of the slave node, \ddot { { \mathbf { u } } } _ { i } ^ { m } are the structural accelerations at the master nodes, and \mathbf { n } ^ { - } \left( \mathbf { x } _ { \mathrm { N } } \right) is the normal vector, pointing into the