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Smart Material Systems and MEMS - Vijay K. Varadan Part 6

Tham khảo tài liệu 'smart material systems and mems - vijay k. varadan part 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 7 Introduction to the Finite Element Method 7.1 INTRODUCTION The behavior of any smart dynamic system is governed by the equilibrium equation Equation 6.49 derived in the last chapter. In addition the obtained displacements field should satisfy the strain-displacement relationship Equation 6.27 and a set of natural and kinematic boundary conditions and initial conditions. Also if the system happens to be a laminated composite with an embedded smart material patch there will be electro-mechanical magnetomechanical coupling introduced through the constitutive model. Obviously these equations can be solved exactly only for a few typical cases and for most problems one has to resort to approximate numerical techniques to solve the governing equations. Equation 6.49 as such is not readily amenable for numerical solutions. Hence one needs alternate statements of equilibrium equations that are more suited for numerical solution. This is normally provided by the variational statement of the problem. Based on variational methods there are two different analysis philosophies one is the displacement-based analysis called the stiffness method where the displacements are treated as primary unknowns and the other is the force-based analysis called the force method where internal forces are treated as primary unknowns. Both these methods split up the given domain into many subdomains elements . In the stiffness method a dis-critized structure is reduced to a kinematically determinate problem and the equilibrium of forces is enforced between the adjacent elements. Since we begin the analysis in terms of displacements enforcement of compatibility of the displacements strains is a non-issue as it will be automatically satisfied. The finite element method falls under this category. In the force method the problem is reduced to a statically determinate struc ture and compatibility of displacements is enforced between adjacent elements. Since the primary unknowns are forces the .