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3.1  Piezoelectric solid finite elements

For 3D solids, the discretized strain and electric fields are linked to the discretized displacement vector (u,v,w) and electric potential φ by:

(2)

where N,x u is a short notation for

(3)

and N_i(x,y,z) are the finite element shape functions. Plugging (2) in (1) leads to the discrete set of equations which are written in the matrix form:

(4)

where {q_mech } contains the mechanical degrees of freedom (3 per node related to u,v,w), and {V } contains the electrical degrees of freedom (1 per node, the electric potential φ). {F_mech } is the vector of applied external mechanical forces, and {Q } is the vector of applied external charges.