An adaptive piecewise weighted integral error correction method for engineering numerical simulation

Abstract

This invention discloses an adaptive piecewise weighted integral error correction method for numerical simulation in engineering, belonging to the field of numerical computation and finite element simulation technology. Traditional integral algorithms employ uniform interval partitioning, globally fixed weights, and global... Lipschitz The constant values ​​are difficult to adapt to the concavity and convexity of functions. This invention uses the sign of the second derivative of the function to determine the concavity and convexity of the interval, and realizes adaptive piecewise segmentation of the integration domain; it solves the local sub-intervals piecewise. Lipschitz constant, based on Hermite‑Hadamard The inequality assigns differentiated weighted correction coefficients to convex and concave regions, and the overall correction error is obtained by accumulating the remainders of piecewise integration. This invention reduces redundant mesh subdivision calculations in smooth regions, significantly reducing the truncation error of strongly nonlinear integration. Under the same piecewise scale, the integration error is reduced by more than 60% compared to traditional algorithms, balancing numerical simulation accuracy and computational efficiency. It is suitable for high-precision engineering numerical calculation scenarios such as structural finite element simulation, numerical solution of heat transfer fields, and integration of deep learning loss functions.

Description

Technical Field

[0001] This invention belongs to the field of numerical computation and finite element engineering simulation technology, specifically relating to an adaptive piecewise weighted integral error correction method. The algorithm module of this invention can be directly embedded into engineering simulation programs, adapting to engineering calculation scenarios such as finite element simulation of mechanical structure stress, numerical solution of heat transfer in heat exchange equipment, integral calculation of machine learning loss functions, and general-purpose engineering numerical approximation calculation software. It can meet the high-precision numerical calculation requirements of multi-field coupled simulation and high-precision integral solution of nonlinear functions. Background Technology

[0002] In engineering numerical simulation work such as finite element analysis, thermo-mechanical coupling simulation, and structural mechanics numerical simulation, definite integral numerical solution is the core underlying computational link, and the truncation error generated by integral solution directly determines the credibility of the overall simulation results. Hermite-Hadamard Inequalities are a classic theoretical tool for analyzing integral truncation errors of nonlinear functions and are now widely used in the design of integral error compensation strategies and the development of numerical integral correction schemes. Currently, integral error correction schemes implemented in engineering fields generally adopt a fixed-step-size uniform interval division and globally fixed-parameter correction approach. However, when dealing with strongly nonlinear engineering functions with alternating concavity and convexity, these schemes have inherent structural defects, which can be summarized in three points: First, the fixed-step, uniformly divided segmentation strategy for the integration interval fails to adapt to the concavity and convexity characteristics represented by the second derivative of the function. In regions with gentle curvature, uniform segmentation generates a large number of redundant sub-intervals, significantly increasing floating-point computation and slowing down simulation iteration. In critical regions with abrupt changes in concavity and curvature, the number of segments is insufficient, making it difficult to capture the abrupt changes in function shape. This leads to a sharp amplification of integration truncation errors, ultimately causing the calculation of key simulation points to fail.

[0003] Second, all integral sub-intervals use a uniform fixed weighting coefficient to complete the error compensation calculation, which cannot achieve differentiated adaptation for the error propagation law of convex function intervals and concave function intervals. Hermite-Hadamard The inequality shows a significant difference in the theoretical upper bound of the error for convex and concave functions. The integral error boundary of concave functions exhibits an inverse distribution characteristic compared to that of convex functions. The globally unified weighted correction mode cannot match the inherent error patterns of the two types of intervals, ultimately leading to an overly conservative estimate of the upper bound of the error. This results in low utilization efficiency of integral error correction and limited gains in accuracy brought about by the correction.

[0004] Third, the overall integration interval relies solely on a single global endpoint. Lipschitz A constant is used as the basis for judging the error boundary. This constant is determined by the maximum value of the second derivative within the entire integration interval, ignoring the differentiated distribution characteristics of the derivative values ​​within each sub-interval. Using a global extremum approach will cause the upper bound of the error in most smooth sub-intervals to be excessively amplified, generating redundant error correction operations and failing to achieve piecewise fine-grained error correction.

[0005] In summary, traditional global fixed-mode integral correction methods have large cumulative integral errors in finite element multi-field coupled simulations and nonlinear engineering function solutions, resulting in simulation outputs that deviate significantly from actual physical conditions. This makes them unsuitable for high-precision numerical calculations required in scenarios such as high-end equipment simulation, precision thermal calculations, and high-precision machine learning model training. Summary of the Invention 1. The technical problem to be solved by the present invention

[0006] Existing numerical integral correction methods suffer from limitations such as uniform fixed segmentation, globally uniform weighted correction coefficients, and globally fixed... Lipschitz To address the shortcomings of constant intervals, such as unreasonable interval division, poor adaptability to concave and convex intervals, conservative error boundary estimation, and insufficient overall integral correction accuracy, this invention proposes an adaptive piecewise weighted integral error correction method for engineering numerical simulations. This invention employs adaptive interval division based on function concavity / convexity, differentiated weighted error compensation for concave and convex intervals, and local sub-interval correction. Lipschitz The three core improvements of independent constant solving enable refined correction of integral errors in nonlinear engineering functions, compressing overall cumulative integral errors while balancing simulation program computational efficiency and integral calculation accuracy.

[0007] An adaptive piecewise weighted integral error correction method for numerical simulation in engineering is characterized by comprising the following execution steps: Step 1: Method initialization, input the nonlinear function of the engineering simulation target. Integral upper and lower limits range Concave-convex segmentation accuracy threshold ; Step 2: Adaptive concave-convex interval segmentation processing Solve for the second derivative of the objective function ,in accordance with The positive and negative signs complete the initial concave-convex sub-interval division; for each divided single concave-convex sub-interval, the amplitude of the second derivative fluctuation within the interval is detected; if the amplitude of the second derivative exceeds a preset precision threshold... Perform bisection subdivision iteration on the current sub-interval until the concavity and convexity of the functions inside all sub-intervals are unique and the fluctuation of the second derivative satisfies the accuracy constraint condition, thus completing the adaptive non-uniform segmentation. Step 3: Solve the local problem sub-interval by sub-interval Lipschitz constant For each subdivided concave-convex sub-region Solve the local problem independently for this subinterval. Lipschitz constant: The traditional scheme with a global uniformity is replaced by piecewise local extremum results. Lipschitz constant; Step 4: Partition Differentiation Weighted Error Correction Based on Concavity / Concavity Attributes Determine the concavity / convexity attribute of the current sub-interval and call the corresponding error compensation formula: If the subinterval is a convex interval, satisfying The piecewise integral error is calculated by calling the weighted error compensation formula specific to the convex interval. If the subinterval is a concave interval, then the following conditions are met: The piecewise integral error is calculated using the weighted error compensation formula specific to the concave interval. Step 5: Superimpose piecewise integration results and summarize errors Calculate the high-precision integral approximation value after introducing the correction term in each sub-interval, and sum the integral results of all sub-intervals to obtain the overall integral correction result; simultaneously sum the upper bound of each segment error to output the upper limit of the global integral total error. Step 6: Output the corrected integral value, segment details, and global error upper bound to complete the entire algorithm calculation.

[0008] Formula for calculating the upper bound of piecewise core error: Let the step size of the sub-interval be:

[0009] Upper bound of piecewise error in convex intervals:

[0010] Upper bound of piecewise error in concave intervals:

[0011] Total global integration error:

[0012] The above-mentioned upper bound of piecewise error is combined with the formula of the remainder term of the composite trapezoidal integral. Hermite-Hadamard The error boundary characteristics of concave and convex functions are derived so that the coefficients of the classical trapezoidal integral remainder term are used in the convex interval, while smaller coefficients are used in the concave interval to complete the fine constraint of the error boundary, thereby realizing the differential error correction between the concave and convex intervals. 2. Beneficial effects

[0013] 1. Adaptive concave-convex segmentation optimizes the interval topology: Based on the concave-convexity of the function, the integration interval is dynamically divided. Compared with the traditional uniform equidistant segmentation, redundant segments are reduced in the smooth region of the function, and the overall number of segments is reduced by more than 40%. This effectively reduces the amount of repetitive calculations in the finite element simulation iteration process and improves the overall simulation speed. 2. Differential weighted correction mechanism for concave and convex intervals: based on convex and concave functions. Hermite-Hadamard Two independent correction formulas are designed for the error theory boundary to solve the problem that the traditional single correction formula cannot adapt to the differentiated error characteristics of the two types of intervals, concave and convex. The error compensation results are consistent with the actual error evolution law of the function. 3. Local Lipschitz Independent solution of constants: The error boundary of each sub-interval is determined by the extremum of its own second derivative, abandoning the conservative value logic of the global maximum value, and the upper bound of the error of the smooth interval is reasonably shrunken, and the fineness of the integral correction is significantly improved. 4. Significant improvement in integral correction accuracy: Under the constraints of the same number of segments and equal hardware computing power, the overall integral cumulative error is reduced by more than 60% compared with the traditional global fixed correction method, and the calculation accuracy of nonlinear engineering function simulation is significantly improved. 5. Strong practicality for engineering applications: The algorithm logic is simple and clear, with a high degree of modularity. It can be directly compiled into a dynamic link library, its function modules can be embedded into mainstream finite element simulation software, and it can be used as an independent numerical calculation program and deep learning framework. It is suitable for diverse industrial scenarios such as mechanical structure stress simulation, pipeline heat transfer simulation, and neural network loss integral solution, and has a clear path for technology transformation and industrialization. Detailed Implementation

[0015] To fully verify the feasibility and error correction performance of the method of this invention, two sets of typical nonlinear test functions for engineering simulation were selected for numerical verification. These correspond to the high-temperature heat transfer simulation condition of a globally convex function and the stress simulation condition of a structure with alternating concave and convex cross-sections, respectively. A control group was set up using a uniform interval trapezoidal integral with the same number of segments + a globally unified function. Lipschitz Comparative experiments were conducted on constant error correction methods. 1. Example 1: Simulation of heat transfer in high-temperature components using a globally convex function.

[0016] Choosing a quartic convex function As a simulation function, corresponding to the simulation model of the temperature field distribution of a high-temperature structure, the integral solution interval is obtained. Segmentation accuracy threshold The theoretical exact value of the definite integral of this function is 48.6.

[0017] Step 1: Solve for the second derivative of the function The second derivative is always greater than 0 within the interval, and the interval as a whole is a single convex attribute interval; Step 2: Perform adaptive bisection subdivision based on the second derivative fluctuation amplitude, ultimately dividing the region into 3 convex sub-intervals: , , ; Step 3: Solve the local problems of the three sub-intervals respectively. Lipschitz The constants are 12, 48, and 108 respectively. Step 4: All sub-intervals are uniformly compensated for integration error using the convex interval-specific error correction formula; Control group (uniformly segmented + globally) Lipschitz Overall integral error (corrected): 12.15; The overall integral error of the adaptive piecewise concavity / convexity correction method of this invention is 4.72. The overall integration error decreased by more than 61%, demonstrating outstanding error correction performance under high-temperature heat transfer integral solving conditions. 2. Example 2: Stress Simulation Case of Alternating Concave-Convex Functions—Variable Cross-Section Structure

[0018] Choosing a sine function As a simulation function, the integration interval is used to simulate the alternating stress distribution of a strain-section beam. The theoretical exact value of the definite integral of this function is 2.

[0019] Step 1: Second derivative , interval The interval is concave if the second derivative is less than 0. The second derivative is greater than 0, indicating a convex interval. The method automatically completes the two concave-convex partitioning. Step 2: Calculate the local values ​​of the concave and convex intervals independently. Lipschitz constant; Step 3: The concave segment uses the concave interval correction formula, and the convex segment uses the convex interval correction formula to achieve differentiated weighted correction; The integral error of the control group (uniformly segmented global correction method) is 0.086. The integral error of the adaptive correction method for concave-convex partitioning in this invention is 0.031. For engineering stress functions with alternating concave and convex characteristics, this method can accurately distinguish the concave and convex properties of intervals and achieve high-precision error correction.

[0020] This method can be encapsulated as an independent computing module, through... C / C++、Python Compile and integrate to Abaqus、ANSYS The finite element simulation preprocessing and solver modules are used for high-precision integral solving of stress fields in engineering machinery structures and numerical simulation of steady-state heat transfer in oil and gas pipelines; they can also be embedded with... PyTorch、TensorFlowA deep learning framework for high-precision solving of definite integrals of complex loss functions; it can also be packaged independently as a desktop numerical computing tool for use in engineering numerical computing projects in universities and research institutes.

Claims

1. An adaptive piecewise weighted integral error correction method for numerical simulation in engineering, characterized in that, Includes the following steps: Step 1: Configure initialization parameters by inputting the nonlinear objective function to be solved, the integration interval, and the accuracy threshold for the concave-convex segmentation. Step 2: Calculate the second derivative of the objective function, divide the initial concave and convex sub-intervals according to the sign of the second derivative, and adaptively divide the sub-intervals into two parts based on the fluctuation amplitude of the second derivative, so that the concave and convex characteristics of each sub-interval are unique and the fluctuation amplitude of the second derivative meets the accuracy threshold requirements. Step 3: For each of the subdivided sub-intervals, calculate the local values ​​of the corresponding sub-intervals. Lipschitz constant; Step 4: Based on the concave and convex characteristics of the sub-intervals, select the appropriate convex interval error correction formula or concave interval error correction formula to complete the solution of the single interval integral error. Step 5: Superimpose the integral calculation results of all sub-intervals to obtain the overall integral approximation value, superimpose the upper bound of each segment error to obtain the global total error, and output the corrected integral value and the global error boundary.

2. The adaptive piecewise weighted integral error correction method for engineering numerical simulation according to claim 1, characterized in that, In step 2, the concavity / convexity of the interval is determined by the sign of the second derivative of the objective function. When the second derivative is always greater than or equal to zero, it is defined as a convex interval, and when the second derivative is always less than zero, it is defined as a concave interval. When the fluctuation amplitude of the second derivative of the sub-interval exceeds the accuracy threshold, the current sub-interval is subdivided by binary iteration until the concavity / convexity of the sub-interval is unique and the derivative fluctuation converges to within the threshold range.

3. The adaptive piecewise weighted integral error correction method for engineering numerical simulation according to claim 1, characterized in that, In step 3, the local Lipschitz The constant is taken as the maximum absolute value of the second derivative of the objective function within the corresponding subinterval, and each subinterval is calculated independently for local values. Lipschitz Constants, not universally uniform Lipschitz The value of the constant.

4. The adaptive piecewise weighted integral error correction method for engineering numerical simulation according to claim 1, characterized in that, In step 4, the upper bound of the first type of error is calculated using the formula based on the remainder of the trapezoidal integral for the convex interval, and the upper bound of the second type of error is calculated using the formula with optimized coefficients for the concave interval. Differentiated weighted error correction is achieved by relying on the error propagation characteristics of the concave and convex intervals.

5. The adaptive piecewise weighted integral error correction method for engineering numerical simulation according to claim 4, characterized in that, The formula for calculating the upper bound of the error in the convex interval is: The formula for calculating the upper bound of the error in the concave interval is: In the formula, represents the local subinterval. Lipschitz The constant is the step size for integration over the subinterval.

6. The adaptive piecewise weighted integral error correction method for engineering numerical simulation according to claim 1, characterized in that, The method is encapsulated as a callable program module, which embeds finite element simulation software, heat transfer numerical calculation program, and deep learning framework for structural stress simulation calculation, thermal field numerical solution, and high-precision integral calculation of neural network loss function.

Images