김경종
82 lines
4.5 KiB
Markdown
82 lines
4.5 KiB
Markdown
---
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type: concept
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title: "Finite Element Method"
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complexity: intermediate
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domain: computational-mechanics
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aliases:
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- FEM
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- finite element analysis
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created: 2026-05-28
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updated: 2026-05-29
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address: c-000006
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tags:
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- concept
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- finite-element-method
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status: current
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related:
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- "[[Computational Mechanics]]"
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- "[[Engineering Mathematical Models]]"
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- "[[Displacement-Based Finite Element Formulation]]"
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- "[[Isoparametric Finite Elements]]"
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- "[[Solid Element Notes]]"
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- "[[Isoparametric Linear Solid Elements]]"
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- "[[Continuum Mechanics Based Four-Node Shell Element]]"
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- "[[On-the-Finite-Element-Analysis-of-Shell-Structures]]"
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- "[[Abaqus Theory Manual]]"
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- "[[Abaqus Element Library]]"
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- "[[A-First-Course-in-the-Finite-Element-Method|A First Course in the Finite Element Method]]"
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- "[[Direct Stiffness Method]]"
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- "[[Finite Element Modeling and Convergence Checks]]"
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sources:
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- "[[Finite Element Procedures]]"
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- "[[A Continuum Mechanics Based Four-Node Shell]]"
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- "[[On-the-Finite-Element-Analysis-of-Shell-Structures]]"
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- "[[Solid Element Notes]]"
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- "[[Abaqus Theory Manual]]"
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- "[[A-First-Course-in-the-Finite-Element-Method|A First Course in the Finite Element Method]]"
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---
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# Finite Element Method
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## Definition
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The finite element method is a numerical procedure for approximating solutions to physical problems by replacing a continuous domain with connected finite elements, choosing interpolation functions over each element, assembling local equations into a global algebraic system, and solving for the unknown field variables.
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## How It Works
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The workflow starts with a physical problem and an idealized mathematical model. The domain is discretized into elements, field variables are approximated using nodal degrees of freedom, element equations are derived from differential, variational, virtual work, or weighted residual statements, and the assembled global system is solved after boundary conditions and constraints are applied.
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## Why It Matters
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Finite element analysis lets engineers approximate complex geometries, material behavior, loads, and boundary conditions that are difficult to solve analytically. The source repeatedly emphasizes that solution quality depends as much on modeling choices as on numerical algorithms.
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The shell element paper adds a focused example: a useful element is not only a mesh topology, but a formulation choice that controls locking, rigid-body behavior, nonlinear kinematics, and benchmark performance.
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The shell FE review reinforces the same modeling-first point: shell results require simultaneous understanding of physical behavior, the selected shell mathematical model, and the finite element approximation.
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[[Solid Element Notes]] adds a compact element-level derivation for 3D continuum elements: natural-coordinate shape functions, Jacobian derivative mapping, `B` and `D` matrices, stiffness integration, and incompatible mode enrichment.
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[[Abaqus Theory Manual]] adds an industrial reference layer: it shows how finite element theory is organized inside a production analysis system through procedures, element libraries, material-point updates, contact, constraints, and coupled-field analyses.
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[[A-First-Course-in-the-Finite-Element-Method|A First Course in the Finite Element Method]] adds a pedagogical layer: it walks the method from springs and bars to trusses, beams, frames, plane elements, axisymmetric elements, isoparametric elements, heat transfer, thermal stress, and dynamics.
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## Key Connections
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- [[Engineering Mathematical Models]] defines what is being solved.
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- [[Displacement-Based Finite Element Formulation]] gives the main solid mechanics derivation.
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- [[Isoparametric Finite Elements]] describes practical element construction.
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- [[Direct Stiffness Method]] shows the basic matrix assembly workflow.
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- [[Finite Element Modeling and Convergence Checks]] captures practical mesh and result checks.
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- [[Continuum Mechanics Based Four-Node Shell Element]] is a focused low-order shell formulation example.
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- [[Static Equilibrium Equation Solvers]], [[Direct Time Integration Methods]], and [[Finite Element Eigenproblem Solvers]] solve the resulting systems.
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- [[Abaqus Element Library]] and [[Abaqus Analysis Procedures]] show how those ideas are packaged in a general-purpose FE code.
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## Sources
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- [[Finite Element Procedures]]
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- [[A Continuum Mechanics Based Four-Node Shell]]
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- [[On-the-Finite-Element-Analysis-of-Shell-Structures]]
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- [[Solid Element Notes]]
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- [[Abaqus Theory Manual]]
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- [[A-First-Course-in-the-Finite-Element-Method|A First Course in the Finite Element Method]]
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