Professor! Today's topic is Lighthill's acoustic analogy, right? What is it?

It's the theoretical foundation for aerodynamic noise generation by turbulent flow. It involves the Lighthill stress tensor and leads to the U^8 law for jet noise.

How is this equation actually solved computationally?

We use spatial discretization with the Finite Element Method (FEM). This involves assembling the element stiffness matrices and constructing the global stiffness equation.

What specifically is meant by matrix solution algorithms?

It refers to solving the system of equations using either direct methods (LU decomposition, Cholesky decomposition) or iterative methods (Conjugate Gradient method, GMRES method). For large-scale problems, preconditioned iterative methods are particularly effective.

The Acoustic Analogy of Lighthill

Category: 解析 | Integrated 2026-04-06

Theory and Physics

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Theoretical foundation of aerodynamic noise generation by turbulent flow. Lighthill stress tensor. Jet noise U^8 law.

Governing Equations

$$ \Box^2\rho' = \frac{\partial^2 T_{ij}}{\partial x_i \partial x_j} $$

$$ T_{ij} = \rho u_i u_j + (p'-c_0^2\rho')\delta_{ij} - \tau_{ij} $$

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Wait wait, so Lighthill's acoustic analogy means it can be used in cases like this too?

Discretization Methods

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How do you actually solve this equation on a computer?

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We use spatial discretization by the Finite Element Method (FEM). We assemble the element stiffness matrix and construct the global stiffness equation.

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We perform transformation to the weak form (variational form) and use formulation by the Galerkin method using test functions and shape functions. The choice of element type (low-order elements vs. high-order elements, full integration vs. reduced integration) directly affects the trade-off between solution accuracy and computational cost.

Matrix Solution Algorithms

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What exactly do you mean by matrix solution algorithms?

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We solve the simultaneous equations using direct methods (LU decomposition, Cholesky decomposition) or iterative methods (CG method, GMRES method). For large-scale problems, preconditioned iterative methods are effective.

Solution Method	Classification	Memory Usage	Applicable Scale
LU decomposition	Direct Method	O(n²)	Small to Medium Scale
Cholesky decomposition	Direct Method (Symmetric Positive Definite)	O(n²)	Small to Medium Scale
PCG Method	Iterative Method	O(n)	Large Scale
GMRES method	Iterative Method	O(n·m)	Large Scale / Non-symmetric
AMG Preconditioner	Preprocessing	O(n)	Very Large Scale

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So if you cut corners on the finite element method part, you'll pay for it later. I'll keep that in mind!

Implementation in Commercial Tools

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So, what software can be used to do Lighthill's acoustic analogy?

Tool Name	Developer/Current	Main File Format
COMSOL Multiphysics	COMSOL AB	.mph
Ansys Mechanical (formerly ANSYS Structural)	Ansys Inc.	.cdb, .rst, .db, .ans, .mac
Ansys Fluent	Ansys Inc.	.cas, .dat, .msh, .jou
Simcenter STAR-CCM+	Siemens Digital Industries Software	.sim, .java, .csv
Abaqus FEA (SIMULIA)	Dassault Systèmes SIMULIA	.inp, .odb, .cae, .sta, .msg

Vendor Lineage and Product Integration History

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Do the origins of each software have some dramatic stories?

COMSOL Multiphysics

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Tell me about "COMSOL Multiphysics"!

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Founded in Sweden in 1986. Started as FEMLAB with MATLAB integration, later renamed to COMSOL. Strong in multiphysics.

Current Affiliation: COMSOL AB

Ansys Mechanical (formerly ANSYS Structural)

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Tell me about "Ansys Mechanical"!

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Developed by Swanson Analysis Systems Inc. (SASI) in 1970. APDL (Ansys Parametric Design Language) based.

Current Affiliation: Ansys Inc.

Ansys Fluent

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Next is about Ansys Fluent. What's the story?

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Developed by Fluent Inc. Acquired by Ansys in 2006. A general-purpose unstructured grid-based CFD solver.