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Finite Element Methods · Chapter 7 of 10 · Advanced

Isoparametric Formulation and Gauss Quadrature

Real meshes are full of distorted, curved elements. The isoparametric idea maps each one back to a perfect parent, and Gauss quadrature integrates the stiffness on that parent with a handful of clever points.

Readiness check

This chapter maps and integrates elements. Tick only what you can do closed-notes.

Recall natural coordinates and parent elements.
Use the chain rule for a change of variable.
Integrate a polynomial over an interval.
Evaluate a function at a given point.
Recall what a Jacobian represents.

0 or 1 weak itemsContinue with this chapter.

2 weak itemsRevisit natural coordinates in Chapter 5.

3 or more weak itemsReview numerical integration in Math: Numerical Methods.

The core idea

An isoparametric element uses the same shape functions to map its geometry and to interpolate its field. The mapping's Jacobian lets the stiffness be integrated on a perfect parent element by Gauss quadrature.

x = Σ N_ix_i, u = Σ N_iu_i (same N)J = dx/dξ relates the coordinates∫ f dx ≈ Σ W_k f(ξ_k) J

A distorted, curved element is hard to integrate directly, so the isoparametric method maps it from a simple parent element, a square or a line on [−1, 1], using the very shape functions that interpolate the displacement. The Jacobian of that mapping, J = dx/dξ in one dimension and a small matrix in two, converts derivatives and lengths between the parent and physical frames, so the stiffness integral can be evaluated entirely on the tidy parent. Those integrals rarely have closed forms, so they are done numerically by Gauss quadrature, which evaluates the integrand at a few special points with chosen weights and is exact for polynomials up to a known degree. Two Gauss points integrate a cubic exactly, which is why so much of FEM rests on just a handful of evaluations per element.

The skill works when: you map to the parent, carry the Jacobian, and choose enough Gauss points for the integrand.

The skill breaks down when: too few Gauss points are used, or the Jacobian is dropped from the integral.

The concept. A perfect square parent element on [−1, 1] maps to a distorted physical quadrilateral. The same shape functions carry both the geometry and the field; the Jacobian relates the two.

The skills, taught in order

Five skills cover the isoparametric idea, the Jacobian, why integration is numerical, Gauss quadrature, and the order of integration.

7.1 The isoparametric concept

An element is isoparametric when the same shape functions describe its geometry, x = Σ N_ix_i, and its field, u = Σ N_iu_i. This lets curved and distorted elements be handled with the same simple functions as a perfect parent.

7.2 The Jacobian

The Jacobian J = dx/dξ (a determinant of a matrix in two and three dimensions) relates derivatives and lengths between the parent and physical coordinates. Derivatives transform as dN/dx = (dN/dξ)/J, and lengths or areas scale by J, so it appears throughout the stiffness integral. A negative or zero Jacobian signals a badly distorted element.

7.3 Why integration is numerical

For a distorted element the stiffness integrand is a complicated function of the natural coordinates, with no convenient closed form. Rather than integrate symbolically, the method evaluates the integral numerically on the parent element, which is fast and general.

7.4 Gauss quadrature

Gauss quadrature approximates an integral as a weighted sum of function values at special points: ∫ f dξ ≈ Σ W_kf(ξ_k). With n points placed optimally, it is exact for polynomials up to degree 2n − 1, so two points integrate a cubic exactly.

Points	Locations ξ	Weights	Exact to degree
1	0	2	1 (linear)
2	±1/√3	1, 1	3 (cubic)
3	0, ±√0.6	8/9, 5/9	5

7.5 The order of integration

The number of Gauss points must match the integrand: too few under-integrates and can cause spurious zero-energy modes, while too many wastes effort. Full integration uses enough points for the exact polynomial; reduced integration deliberately uses fewer to soften an over-stiff element, a deliberate trade.

Engineering connection: isoparametric quadrilaterals and hexahedra, integrated by Gauss quadrature, are the standard elements in every commercial FEA code.

Worked example 1: the Jacobian of a mapped element

A one-dimensional element maps the parent coordinate ξ ∈ [−1, 1] onto the physical span x = 3 to x = 7. Find the Jacobian of the mapping and use it to relate a physical integral to a parent integral.

Figure 1. The parent length of 2 maps to a physical length of 4, so the Jacobian is 2. Every physical integral picks up this factor.

ProblemFind the Jacobian for the mapping in Figure 1.
Given / findx₁ = 3, x₂ = 7, parent ξ ∈ [−1, 1]. Find J = dx/dξ.
AssumptionsLinear two-node isoparametric mapping.
ModelWith x = N₁x₁ + N₂x₂ and N = (1 ∓ ξ)/2, the Jacobian is dx/dξ = (x₂ − x₁)/2.
EquationsJ = dx/dξ = (x₂ − x₁)/2∫ f dx = ∫ f J dξ
SolveJ = (7 − 3)/2 = 2. A physical integral becomes ∫ f dx = ∫ f × 2 dξ over the parent.
CheckThe parent interval has length 2 and the physical element length 4, so the stretch is 4/2 = 2, matching the Jacobian. A derivative in x is (1/J) times the derivative in ξ.
ConclusionThe Jacobian carries the element's size and shape into the parent integral. In two and three dimensions it becomes a matrix whose determinant plays the same role.

Result. J = 2, so ∫ f dx = 2 ∫ f dξ.

Worked example 2: Gauss quadrature

Integrate f(ξ) = ξ² + 1 over the parent interval [−1, 1] using two-point Gauss quadrature (points ±1/√3, weights 1 and 1), and compare with the exact value and with one-point quadrature.

Figure 2. Two Gauss points sample the parabola where its average lives, giving the exact integral. One point at the centre misses the curvature and under-integrates.

ProblemIntegrate ξ² + 1 on [−1, 1] by two-point Gauss quadrature in Figure 2.
Given / findf(ξ) = ξ² + 1; two points ±1/√3 with weights 1, 1. Find the quadrature value and compare.
AssumptionsGauss-Legendre quadrature on the standard interval.
Model∫ f dξ ≈ W₁f(ξ₁) + W₂f(ξ₂); the exact integral is found analytically for comparison.
Equations∫₋₁¹(ξ² + 1)dξ = [ξ³/3 + ξ] = 8/32-pt: f(−1/√3) + f(1/√3)
SolveEach point gives f = (1/3) + 1 = 4/3, so the sum is 4/3 + 4/3 = 8/3 ≈ 2.667, equal to the exact value. One-point quadrature gives 2 × f(0) = 2 × 1 = 2, which is too low.
CheckTwo points are exact for any cubic, and ξ² + 1 is only quadratic, so exactness is expected. One point is exact only for linear functions, so it misses the ξ² term.
ConclusionGauss quadrature integrates the element stiffness with a few points, and the number of points must match the polynomial degree of the integrand.

Result. Two-point Gauss gives 8/3, exact; one point gives 2, too low.

Misconceptions and diagnostics

Mistake	Symptom	Diagnostic question	Correction
Dropping the Jacobian	Integral off by the element scale	"Did I multiply by J?"	Carry the Jacobian into every parent integral.
Too few Gauss points	Under-integration, zero-energy modes	"Is the order enough for this polynomial?"	Use n points for degree up to 2n − 1.
Badly distorted element	Negative or zero Jacobian	"Is the element shape acceptable?"	Improve the mesh so J stays positive.
Over-integration as a fix	Slow runs with no accuracy gain	"Do I need more points than the degree?"	Match the order; do not over-integrate by habit.

Practice ladder

Level 1 · Direct skill

An element maps ξ ∈ [−1, 1] to x = 0 to x = 10. Find the Jacobian.

Show answer

J = (10 − 0)/2 = 5.

Level 2 · Mixed concept

Integrate f(ξ) = 3ξ² over [−1, 1] by two-point Gauss quadrature.

Show answer

f(±1/√3) = 3 × (1/3) = 1 each, so the sum is 1 + 1 = 2. Exact: ∫3ξ²dξ = [ξ³] = 2. Exact.

Level 3 · Independent problem

How many Gauss points are needed to integrate a degree-5 polynomial exactly, and where do two of them sit?

Show answer

Three points (since 2n − 1 = 5 gives n = 3), at ξ = 0 and ξ = ±√0.6, with weights 8/9 and 5/9.

Level 4 · Transfer to real engineering

Explain why distorted elements degrade accuracy, in terms of the Jacobian and the mapping.

What good work looks like

The idea that severe distortion makes the Jacobian vary or approach zero, so the mapping and its derivatives become inaccurate, degrading the stiffness integral.

Working with AI, and proving it yourself

Use AI as an examiner, not a solver

"Check that I used enough Gauss points for this integrand's degree."

"Give me three integrands; I will say how many Gauss points each needs."

"Integrate this element for me." Doing the quadrature yourself is the skill.

"What is the Jacobian?" Differentiating the mapping is the point.

Portfolio task

Set up an isoparametric element: write the mapping, compute the Jacobian, and integrate a sample stiffness term by Gauss quadrature with the correct number of points.

Must include: a Jacobian, a Gauss-point choice justified by the degree, and an integrated value.

Retrieval and spaced review

Closed notes. Answer out loud, then reveal.

1. What makes an element isoparametric?

The same shape functions describe its geometry and its field.

2. What does the Jacobian do?

Relates derivatives and lengths between the parent and physical coordinates.

3. Why integrate numerically?

The distorted-element integrand has no convenient closed form.

4. How exact is n-point Gauss quadrature?

Exact for polynomials up to degree 2n − 1.

5. What does a zero Jacobian signal?

A badly distorted, invalid element.

TodayFinish this quiz and Levels 1 and 2 of the ladder.

+1 dayRe-derive the Jacobian and the quadrature from a blank page.

+3 daysIntegrate two new functions by Gauss quadrature.

+7 daysCarry the method into heat transfer, Chapter 8.

+30 daysReuse Gauss quadrature when reasoning about element accuracy.

Textbook mapping

Item	Mapping
Primary source	Logan, A First Course in the Finite Element Method, Chapter 10 (Isoparametric Formulation)
Cross-reference	Reddy, Ch. 9 · Math: Numerical Methods
Core topics	7.1 Isoparametric concept · 7.2 Jacobian · 7.3 Numerical integration · 7.4 Gauss quadrature · 7.5 Order of integration
Engineering connection	Isoparametric elements with Gauss integration are the standard in commercial codes.
Read next	Chapter 8: Finite Elements for Heat Transfer.