CECM: A continuous empirical cubature method with application to the dimensional hyperreduction of parameterized finite element models (2308.03877v1)

Abstract: We present the Continuous Empirical Cubature Method (CECM), a novel algorithm for empirically devising efficient integration rules. The CECM aims to improve existing cubature methods by producing rules that are close to the optimal, featuring far less points than the number of functions to integrate. The CECM consists on a two-stage strategy. First, a point selection strategy is applied for obtaining an initial approximation to the cubature rule, featuring as many points as functions to integrate. The second stage consists in a sparsification strategy in which, alongside the indexes and corresponding weights, the spatial coordinates of the points are also considered as design variables. The positions of the initially selected points are changed to render their associated weights to zero, and in this way, the minimum number of points is achieved. Although originally conceived within the framework of hyper-reduced order models (HROMs), we present the method's formulation in terms of generic vector-valued functions, thereby accentuating its versatility across various problem domains. To demonstrate the extensive applicability of the method, we conduct numerical validations using univariate and multivariate Lagrange polynomials. In these cases, we show the method's capacity to retrieve the optimal Gaussian rule. We also asses the method for an arbitrary exponential-sinusoidal function in a 3D domain, and finally consider an example of the application of the method to the hyperreduction of a multiscale finite element model, showcasing notable computational performance gains. A secondary contribution of the current paper is the Sequential Randomized SVD (SRSVD) approach for computing the Singular Value Decomposition (SVD) in a column-partitioned format. The SRSVD is particularly advantageous when matrix sizes approach memory limitations.

• The paper presents the Continuous Empirical Cubature Method (CECM), an algorithm designed for efficient integration rules in parameterized finite element models, crucial for hyper-reduced order models.
• CECM employs a two-stage process: an initial interpolatory rule generation using Discrete Empirical Cubature Method (DECM) followed by an iterative sparsification stage to minimize integration points while controlling error.
• Numerical tests demonstrate CECM's effectiveness, significantly reducing integration points (e.g., from 42,471 to 6 with <0.005% error in one case) while maintaining high accuracy, applicable across various model types.

Continuous Empirical Cubature Method for Integration of Parameterized Finite Element Models

The paper presents the Continuous Empirical Cubature Method (CECM), an algorithm developed to devise efficient integration rules for parameterized finite element models. The CECM aims to reduce the number of integration points used in traditional cubature methods, achieving computational efficiency while maintaining accuracy. This approach is particularly beneficial in hyper-reduced order models (HROMs), where reducing computational cost is critical for large-scale simulations.

Methodology

The CECM is structured as a two-stage process. In the first stage, an initial approximation of the cubature rule is generated using a point selection strategy. This approach applies the Discrete Empirical Cubature Method (DECM), which selects a number of points equivalent to the number of functions being integrated. The DECM part of the method ensures that the initial cubature rule is an interpolatory rule.

The second stage involves a sparsification strategy. Here, the algorithm iteratively reduces the number of integration points by adjusting the spatial coordinates of the points and recalculating the weights associated with the remaining points. This optimization process aims to set as many weights as possible to zero, minimizing the integration points while ensuring the integration error remains within a pre-defined threshold.

Numerical Results and Claims

The paper illustrates the CECM's effectiveness through several numerical tests:

• For univariate and multivariate Lagrange polynomials, the CECM retrieves optimal Gaussian integration rules, particularly for polynomials of odd degree.
• The CECM's ability to reduce the integration points significantly is demonstrated in a multiscale finite element model application. For instance, a complex model originally requiring 42,471 points is reduced to a mere 6 points using the proposed method, maintaining an error below 0.005%.
• Another notable example concerns the efficient cubature of an exponential-sinusoidal function within a 3D domain, reducing the number of required points from 133 to 38.

These strong numerical results underscore the CECM's potential for significant computational savings without sacrificing accuracy.

Theoretical and Practical Implications

The theoretical underpinnings of the CECM leverage the orthogonal basis provided by Singular Value Decomposition (SVD) to achieve sparsity in integration rules. The CECM's practical implications are profound in fields that perform large-scale simulations, such as structural engineering, fluid dynamics, or any domain utilizing finite element analysis.

Additionally, the paper introduces the Sequential Randomized SVD (SRSVD), which facilitates the computation of SVD for large matrices typical of finite element analyses. This method aligns with current trends towards data-driven and high-performance computational techniques, allowing large problems to fit within typical memory constraints of workstation-class hardware.

Future Developments and Impact

The development of the CECM paves the way for further advancements in computational efficiency for high-dimensional problems and models with complex parameter spaces. Future research could explore extensions of this method to non-Gaussian rules or adapt it to other forms of integrals found in quantum computations or statistical physics.

Moreover, the focus on reducing computational load aligns well with ongoing efforts in computational science to address energy efficiency and environmental sustainability by optimizing simulations requiring substantial computational resources.

In conclusion, the CECM represents a significant advancement in the field of numerical integration for parameterized models, offering both theoretical innovation and practical utility. It holds exciting possibilities for reshaping computational approaches in various scientific and engineering applications.