he did a marvelous thing by creating the field of differential equations by allowing us to reformulate the problems of natural sciences to geometrical problems of curves in plane and surfaces in space.
The solution to physical problems is curve in plane or surface in space, geometry.
Departure from classical Rayleigh Ritz Method leads to FEM. The two departures are:Trial functions are defined for sub-domains.Generally, the use of values of space variable at the nodes as unknowns.
Rayleigh Ritz Method --> Finite Element Method
Classical Rayleigh Ritz Method is a method of finding displacements at various nodes based on the theorem of minimum potential energy.
Rayleigh Ritz Method
This concept of minimum total potential energy is nothing but can be called as the special case of the virtual work principle.
virtual work principle-->minimum total potential ennergy
Principal of Virtual work and minimum PE is a particular form of Variational Procedure only.
only variational Procedure uses virtual work and minimum PE?
其中重复指标N意味着求和， uN(e)u_N ^{(e)}u_N ^{(e)} 表示子域(element)节点N上的值，并且 ΦN(e)\Phi_{N}^{(e)}\Phi_{N}^{(e)} 称为子域(element)试探函数(trial functions ),或称为插值函数(interpolation functions)、形状函数( shape functions)或基函数( basis functions).
FEM插值函数
假设变量 u(e)(x)u ^{(e)} (x)u ^{(e)} (x) 是 xxx 的线性函数
为什么是坐标的线性函数？