Capacitance matrix

Above, Afj is the capacitance matrix, is the conductivity matrix, is the stiffness matrix. Be (0,l) is a parameter. 

A way of eliminating the domain integral in Eq. 8.81 is based on the well-known idea of combining a particular solution to a complementary solution of a homogeneous problem. This idea was discussed in Buzbee et al. (1971), who named the method the capacitance matrix method. Brebbia and Nardini (1986) have applied the concept to boundary element method and called it the dual reciprocity method (DRM). Henry and Banerjee (1988) have also proposed a related method in the name of the particular integral method. Now, let us consider a particular solution T that satisfies... 

Now the concept of the capacitance matrix as introduced already in (4.6), (4.7) is brought into play be writing the densities u and c. as functions of the potentials. 

Eq. (7.19) expresses that the matrix ap./acj is symmetric. Since C.j is the inverse of ay./ac. we have thus shown that the capacitance matrix C.j. is symmetric ... 

Now, the condition dL/dt 0 as required for the Liapunov-stability of the equilibrium is reduced to the condition that the matrix of the linear phenomenological coefficients is positive definite. This latter property, however, is a direct consequence of the second law of thermodynamics as we have shown in (3.82). With this conclusion we have reconfirmed our preliminary result of the preceding section in a formally precise way a sufficient condition for the stability of the equilibrium is a) a positive-definite capacitance matrix such that L 0 and b) the second law of thermodynamics such that dL/dt 0. Let us emphasize once more the significance of the equivalence between dL/dt < 0 and the second law in the form of (3.82). This equivalence, however, is valid only in the range of validity of the linear relations in (3.81). If the fluxes I were some nonlinear functions of the forces F as will be the case in situations far from the thermodynamic equilibrium, dL/dt 0 is no longer guaranteed by the second law and possibly may no longer be valid at all. 

It should be emphasized that the assumption L 0 is far from being trivial or unproblematic but implies a positive capacitance matrix for all possible steady states or, in other words, stable equilibrium states under any external equilibrium conditions for the system. By the expression "equilibrium conditions" we mean external conditions under which the system is capable of realizing an equilibrium state. The assumption L 0 therefore excludes for example any kind of phase transitions. 

Frequently the stability criterion of Glandorff and Prigogine is expressed as an evolution criterion. By interpreting the variations and 6c. in (7.24) as variations during a real time evolution of the system we can formulate the assumption of a positive definite capacitance matrix equivalently as... 

The other class of formulations of the FEM is based on the definition of an effective specific heat. This results in the inclusion of the latent heat effect in the capacitance matrix. There are a number of ways in which this can be provided for. Each of these methods makes use of an enthalpy temperature curve, for example. 

Ri, Li, and Q are the positive sequence components of the line resistance, inductance, and capacitance matrix... 

The capacitance matrix looks similar to the impedance matrix in Table 3.3a-2. The capacitance between the core and sheath of the homogeneous model is identical to that of the solidly bonded cable. The equivalent capacitance Q4 of the cross-bonded cable in Table 3.3b-2 is given as the sum of the elements as shown in Equations 3.44 and 3.45. 

Kolmogorov second order velocity structure function (m /s ) Capacitance matrix... 

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