Intuition The One Core IdeaA big structure is just many small springs glued together at shared points. Assembling the global stiffness matrix means writing one giant "spring law" for the whole structure by adding up each little spring's stiffness wherever two springs share a point. Everything on this page exists to make that one sentence precise.
Before you can read the parent note Assembling Global Stiffness , you must be able to read every squiggle in it without stopping to guess . This page introduces each symbol from absolute zero: what it means in plain words, what picture it stands for, and why the topic cannot do without it. Read top to bottom — each idea leans on the one above it.
Push on a spring, it pushes back. Push twice as hard, it stretches twice as far (for small pushes). That "how hard it pushes back per unit stretch" is stiffness . A whole spacecraft bracket behaves like thousands of tiny springs wired together — that is the entire mental model.
Look at the figure. On the left, a single spring: pull the right end by an amount u F F = k u slope k finite element mesh .
k "How much force do I get back per unit of stretch?" Units: newtons per metre (N/m k k
This is the whole idea behind Element Stiffness Matrices and behind Finite Element Method Overview : replace a continuous solid with a network of these little stiffnesses.
u == u == displacement ) is the standard symbol. If node 2 slides 0.3 mm u = 0.3 mm
But a point in space can move in more than one way. That is what a degree of freedom captures.
The figure shows the three cases side by side. The little arrows are the DOFs — each arrow is one number the solver must find.
Intuition Why we count DOFs so carefullyEvery DOF is one unknown number the computer must solve for. If your mesh has N N N N K
The symbol u x x
The symbol θ z z
We have many DOFs, so we stack their numbers into lists and grids.
Definition Vector (bold lowercase, e.g.u u all the displacement unknowns of the whole mesh, one slot per DOF. F
Definition Matrix (bold uppercase, e.g.K machine that turns one vector into another. Feeding a displacement vector into K Ku = F
Ku = F F = k u K u F F u
How matrix × vector works (you must be able to read this): row i K u i
F i = ∑ j K ij u j
∑ j ∑ K ij u j j K ij u j
K ij i row , j column . So K 23 the force appearing at DOF 2 when DOF 3 is displaced by one unit and everything else is held fixed.
K ij i j
Each element (each little spring/bar) is easiest to describe along its own axis . But the whole structure lives in one shared frame. Two vocabularies:
Definition Local coordinatesThe element's private frame, usually with one axis pointing along the bar. In this frame the bar's stiffness is simple (see the L E A
Definition Global coordinatesThe single shared x y z
Definition The superscript( e ) k ( e ) e k ( e ) element's stiffness grid. Uppercase K whole structure's stiffness grid. The bracket ( e ) not a power — you never multiply by it.
k ( e ) k e e k ( 1 ) k ( 2 )
The job of translating from a local frame to the global frame is handled by Coordinate Transformations in FEM ; the little numbers E A L k ( e ) Element Stiffness Matrices .
These three letters set how stiff a single bar is.
E material resists stretching. Steel has large E Pa = N/m 2
A thick the bar is (area of its cut face). A fatter bar is stiffer. Units: m 2
L long the bar is. A longer bar stretches more for the same force, so it is less stiff. Units: m
The figure shows how L E A ⇒ ⇒
Definition Connectivity (topology)A little table saying "element e It is how the computer knows where to drop each element's numbers into the big grid.
In the figure, element 1 connects global nodes 1 and 2; element 2 connects nodes 2 and 3. Node 2 is shared . That sharing is the whole reason we add contributions — and why K 22
Intuition Why connectivity is the key to assemblyAssembly = "each element hands its small stiffness numbers to the big matrix, filed under the global DOF numbers it touches." Without the connectivity table you would not know which rows and columns to file them under. The mapping "local DOF → Boolean localization matrix L ( e ) picks out the right slots.
The role of L ( e ) e
Three final vocabulary items the parent leans on:
The practical assembly move: take each small entry k ij ( e ) add it onto the big-matrix slot K g i , g j
K ij = K j i i j j i
Almost all entries of K same element . Storing only the non-zeros is the subject of Sparse Matrix Storage , and it decides whether you use Direct vs Iterative Solvers .
Definition Positive semi-definiteThe energy phrase: u T Ku ≥ 0 u negative strain energy. It equals zero only for rigid-body motions (whole structure drifts without stretching). Pinning down some DOFs — Boundary Conditions in FEM — removes those free drifts and makes K
"Sparse" means ::: nearly all matrix entries are zero.
A rigid-body mode is ::: a motion that moves the whole structure without stretching any element (zero strain energy).
Assemble global stiffness
Properties symmetric sparse PSD
Each box is one thing this page defined; the arrows show what feeds what. The bottom box is the parent topic.
Test yourself — cover the right side and answer before revealing.
Meaning of stiffness k
Meaning of a DOF ::: one independent way a node can move (or rotate).
How many DOFs a free 3D node has ::: 6 (three translations, three rotations).
What u F
What Ku = F F = k u
What the subscript in K ij i j i j
What the ∑
What ( e ) k ( e ) e
Difference between local and global coordinates ::: local = along the element's own axis; global = the one shared frame of the whole structure.
What E A L
Value of a single bar's stiffness ::: E A / L
What connectivity tells you ::: which global nodes each element joins.
What scatter-add does ::: adds each element's small entries into the global matrix at the right global DOF slots.
Why K 22
What "sparse" means ::: almost every entry is zero.
What positive semi-definite means physically ::: strain energy is never negative; zero only for rigid-body motion.
