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algorithm analysis analytical applied approximately aqueduct assumed axial beam behavior Beskos boundary conditions boundary element method calculated cell ceramic coefficients components composite computational concrete constant constitutive equations convection coordinate crack damage deformation developed direction dislocation density displacement displacement vector distribution domain dynamic effect elastic energy Engineering equation experimental Figure finite element analysis finite element method flip-chip flow force formulation frequency functions heat impact initial integral interface joint laminate layer linear load material properties Mechanics membrane mesh microcrack modulus nodal nodes nonlinear numerical simulation obtained parallel parameters penetration performed plane plastic plate Poisson's ratio predicted present pressure problem processors projectile ratio reinforced residual stress rotation shear stress shown in Fig solder solid solution solved steel step stiffness matrix structure surface symmetric target technique temperature tensor thermal thickness Trans transverse values vector velocity Verti vibration weld Young's modulus Zapotec