有限要素法

Finite element method, FEM

[目次：理論(電子の散乱／回折／結像)]

有限要素法とは、解析的に解くことが難しい偏微分方程式の近似解を得る数値解析法の1つ。注目する物体を単純な形状をした有限の大きさの要素に分割し、各要素の物理量(温度や応力など)をできるだけ簡単な方程式で近似し、それらの連立方程式を立てる。こうして得られた連立方程式を、各要素の表面での物理量を境界条件として解くことで、物体全体にわたる物理量の分布を求める方法である。物体を多面体に細分化するため、複雑な形状の物体に適用しやすい。電子顕微鏡においては、機械的強度や熱分布の計算、磁界レンズや静電レンズの磁場および静電場分布の計算などに利用されている。レンズのポールピース開発では、有限要素法で求めた磁場分布を用いて電子軌道を計算することで収差係数を求め、磁極形状の最適化設計を行う。

"Finite element method (FEM)" is one of numerical analytical methods to obtain an approximate solution of partial differential equations that are difficult to solve analytically. First, an object of interest is divided into elements that each has a simple shape and a finite size. Next, physical quantities (temperature, stress, etc.) of each element are approximated by a simpler equation, and then the equations for the elements are combined to construct simultaneous equations. By solving the obtained simultaneous equation under the boundary conditions of the physical quantities at surfaces of the elements, the distribution of the physical quantities over the object are obtained. Since an object is subdivided to polyhedrons, FEM can be conveniently applied to complicated-shape objects. In electron microscopy, the method is used for calculation of mechanical strength and thermal distribution, calculation of distributions of magnetic fields and electrostatic fields of magnetic lenses and electrostatic lenses, etc. In the development of lens polepieces, aberration coefficients are obtained by calculation of electron trajectories using the magnetic field distributions obtained by FEM, and then the shapes of magnetic poles are optimized.