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An efficient algorithm for evaluating Green's functions of discretized elasticity system

This paper proposes a novel, computationally efficient algorithm for constructing discrete Green's functions in two-dimensional linear elasticity systems, demonstrating its validity and applicability to methods like the finite element method through numerical analysis and examples.

Original authors: Jingcheng Miao, Gang Pang, Shaoqiang Tang

Published 2026-08-27

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CC BY 4.0

Original authors: Jingcheng Miao, Gang Pang, Shaoqiang Tang

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a vast, solid sheet of material, like a thick pane of glass or a block of steel. When you push on a single point of that sheet, the entire structure responds. The material stretches, compresses, and shifts in a complex pattern that ripples outward from your finger. Engineers and scientists need to predict exactly how this happens to design safe bridges, durable aircraft, and reliable microchips. The mathematical tool used to describe this reaction is called a Green's function. Think of it as a master key: if you know how the material reacts to a single, tiny push at one specific spot, you can mathematically combine that knowledge to figure out how it reacts to any complex force, no matter how large or complicated. For decades, calculating this master key for a computer simulation has been a slow, grinding process, often requiring the computer to solve the problem from scratch for every single point in the material.

A team of researchers from Peking University and Beihang University has developed a new way to generate these master keys much faster. Their work focuses on a specific type of material model known as linear elasticity, which describes how solid objects deform under stress. In their approach, they break the continuous material down into a grid of tiny points, a standard technique in computer modeling. Instead of trying to solve the equations for the entire grid all at once, which is computationally expensive, they first solved a simplified version of the problem where the material wraps around on itself like a video game world, with no edges. This "periodic" version is easy to solve quickly using a mathematical shortcut known as the fast Fourier transform, which efficiently processes patterns in data.

Once they had this quick solution for the wrapping-around version, the researchers used a clever mathematical trick to adjust it for a real, finite block of material with distinct edges. They treated the difference between the wrapping version and the real version as a set of corrections that could be calculated by looking only at the boundary nodes, the points along the edge of the grid. By combining the fast periodic solution with these boundary corrections, they constructed the complete Green's function for the actual material. This method allowed them to generate all the necessary response data for a grid of 129 by 129 points in just 12.8 seconds on a standard personal computer. In contrast, the traditional method of solving the problem directly for every single point took over two minutes for the same grid, and the time gap widened significantly as the grid size increased.

The researchers tested their algorithm not only on simple grids but also on more complex scenarios involving cracks in the material and different types of boundary conditions, such as fixed edges or applied forces. In every case, the results matched the accuracy of the traditional, slower methods. They demonstrated that this new approach works for both finite difference schemes, which use simple grid points, and finite element methods, which are often used for irregular shapes. The study confirms that by leveraging the speed of periodic calculations and applying a linear combination to handle the boundaries, it is possible to solve complex elasticity problems with a fraction of the usual computational cost. This efficiency opens the door for more detailed simulations of material behavior, particularly in fields like fracture mechanics, where understanding how cracks propagate requires solving these equations repeatedly and with high precision.

Technical Summary: An Efficient Algorithm for Evaluating Green's Functions of Discretized Elasticity System

Problem Statement

The paper addresses the computational challenge of evaluating Green's functions for fully discretized two-dimensional linear elasticity systems on finite rectangular domains. While Green's functions are fundamental tools for solving partial differential equations (PDEs) in mechanics and engineering, their accurate computation on finite domains is difficult. Traditional approaches often rely on continuous Green's functions on infinite domains combined with composite region strategies, which introduce singularities and accuracy issues when applied to finite domains. Alternatively, directly solving the discretized system for every possible source point (Direct Numerical Simulation or DNS) is computationally expensive, scaling poorly as the mesh size increases. The authors aim to develop an algorithm that efficiently constructs discrete Green's functions for both finite difference and finite element method (FEM) schemes, enabling exact solutions for the specific discretization used.

Methodology

The proposed algorithm constructs the discrete Green's function for a rectangular domain by leveraging periodic discrete Green's functions and the Fast Fourier Transform (FFT). The methodology proceeds through the following logical steps:

Discretization and Transformation:
The continuous elasticity equations are transformed using an exponential factor $e^{\alpha x + \alpha y}$ to ensure solution uniqueness for small $\alpha$ . The domain is discretized using a uniform grid. The authors define a central difference scheme for the finite difference method and a bilinear piecewise continuous interpolation for the FEM.
Periodic Green's Functions via FFT:
Instead of solving the system directly on the finite domain, the algorithm first computes periodic Green's functions on the entire discrete space. These are obtained efficiently using the Discrete Fourier Transform (DFT). The Fourier coefficients are derived analytically from the discrete operators, allowing the periodic response to a point source to be calculated via inverse FFT.
Decomposition into Interior and Boundary Components:
The core of the method involves decomposing the solution on the finite rectangular domain into:
- Interior Green's Functions: Responses to interior sources with zero boundary values.
- Boundary Green's Functions: Responses to boundary sources (displacement or traction) with zero interior sources.
Linear Combination Strategy:
The authors establish a theoretical framework (via Lemmas 3 and 4 and Theorem 3) showing that the interior and boundary Green's functions on the finite domain can be expressed as linear combinations of the periodic Green's functions. Specifically:
- The boundary Green's functions are represented by linear combinations of the interior Green's functions.
- The interior Green's functions are represented by linear combinations of the periodic Green's functions.
- This relationship allows the construction of the finite domain Green's functions by solving a linear system involving the boundary nodes, rather than solving the full system for every interior source point.
Algorithm Flow:
The algorithm (detailed in Box I) computes the Fourier coefficients, generates periodic Green's functions via FFT, solves for the transformation coefficients between periodic, interior, and boundary sets, and finally constructs the required Green's functions. This approach is applicable to both the central difference scheme and the FEM, where the stiffness matrix rows replace the finite difference operators.

Key Contributions

Efficient Construction Algorithm: The paper proposes a novel algorithm that significantly reduces computational cost compared to Direct Numerical Simulation (DNS). By utilizing FFT and linear combination strategies, it avoids the need to solve the full system for every source point individually.
Extension to FEM: The method is not limited to finite difference schemes; it is explicitly extended to the Finite Element Method, handling mixed boundary conditions (Dirichlet and Neumann) effectively.
Theoretical Framework: The authors provide rigorous proofs (Theorems 1–3) establishing the existence and uniqueness of the discrete Green's functions and the linear relationships between periodic, interior, and boundary components.
Vector Parallelism: The algorithm is designed to exploit vector parallelism, further reducing time costs, particularly for large-scale problems.

Results

Numerical experiments were conducted on a personal computer using MATLAB with vector parallelism.

Computational Efficiency: Table 1 compares the computational time of the proposed algorithm against DNS for various mesh sizes (ranging from $9 \times 9$ $9 \times 9$ to $129 \times 129$ $129 \times 129$ ).
- For a $129 \times 129$ mesh, the proposed algorithm generated all 16,641 interior Green's functions in 12.8 seconds.
- In contrast, DNS required 141 seconds for the same task.
- The time advantage of the proposed method increases as the mesh scale grows.
Accuracy: The $L_2$ error analysis (Figure 3) demonstrates that the method achieves second-order accuracy, consistent with the central difference scheme. The error is measured against reference solutions computed on a much finer $1025 \times 1025$ mesh.
Application to Fracture: The algorithm was applied to a heterogeneous elasticity system modeling a single-edge notched specimen with a crack. The displacement and strain fields obtained using the proposed Green's function-based iterative solver were found to be nearly identical to those obtained by directly solving the global stiffness matrix system (FEM), validating the method's accuracy for complex geometries and boundary conditions.

Significance and Claims

The paper claims that the proposed algorithm offers a significant reduction in computational time compared to traditional DNS for evaluating discrete Green's functions, particularly for large grids. The authors emphasize that:

The method provides exact solutions for the specific finite difference or finite element scheme employed, avoiding the errors associated with approximating finite domain problems with infinite domain Green's functions.
The approach is generalizable to other partial/ordinary differential equations and can be extended to three-dimensional cases.
The availability of efficient discrete Green's functions facilitates advanced applications, such as clustering analysis methods and homogenization of materials with fractures, where interaction tensors derived from Green's functions are critical.
The method supports iterative algorithms for solving non-linear problems, as demonstrated by its application to the heterogeneous elasticity system.

The authors maintain a modest tone, noting that while the current work focuses on two-dimensional equilibrium equations, the framework is designed to be extensible. They explicitly state that the detailed discussion of discrete Green's functions for FEM in the context of cluster analysis will be presented in a forthcoming paper.