Equation Solver Variable Elimination for Simulation Efficiency
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Solution Overview
Problem
Solving large, multi-variable systems of equations consumes significant computational resources such as memory and processing time, particularly in complex simulations like hydraulic fracturing, making it challenging to obtain high-fidelity results in a timely manner with reasonable hardware.
Innovation Solution
An efficient equation solver reduces the number of variables by identifying and eliminating derived variables, using storage policies to further reduce the number of variables, and iteratively solving for primitive variables, thereby improving memory usage and computational efficiency.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If the number of variables in the system of equations is increased to model complex hydraulic fracturing scenarios, then the fidelity and accuracy of the simulation results are improved, but the computational resources (memory and processing time) required to solve the system increase significantly
Solution Approach 1:
The patent extracts and eliminates derived variables from the system of equations, keeping only the essential primitive variables. This is achieved by identifying variables that can be expressed in terms of others (derived variables) and removing them from the system, thereby reducing the total number of variables while preserving the essential physics of the hydraulic fracturing simulation.
Solution Approach 2:
The patent changes the parameter representation by transforming the system from using all original variables to using a reduced set of primitive variables. This parameter transformation allows the same physical system to be represented with fewer degrees of freedom, reducing memory requirements and computational complexity while maintaining simulation accuracy.
2Productivity
If efficient solver algorithms are used to reduce computational time, then the speed of obtaining solutions is improved, but the memory resources required still remain significant
Solution Approach 1:
The patent extracts derived variables that consume memory resources but do not need to be stored as independent unknowns. By eliminating these variables through algebraic substitution and expressing them in terms of primitive variables, the system reduces its memory footprint while allowing efficient solvers to work with a smaller, more manageable set of equations.
3Measurement precision
If the system of equations is solved with all variables included to ensure complete accuracy, then the precision of the solution is improved, but the computational time required increases significantly
Solution Approach 1:
The patent removes derived variables from the system that would otherwise increase computational time. These variables are not completely eliminated from the physics but are instead expressed algebraically in terms of primitive variables, allowing the solver to work with a reduced system that converges faster while still capturing the full physical behavior when the derived variables are computed from the solution.
Solution Approach 2:
The patent performs preliminary algebraic manipulation to express derived variables in terms of primitive variables before the numerical solution process begins. This pre-processing step eliminates the need for the solver to handle derived variables as independent unknowns, reducing the computational burden of each iteration while preserving the ability to compute accurate derived quantities from the final solution.
Data Source
AI summary
To improve efficiency for solving a system of equations, art equation solver uses variable reduction techniques to reduce a number of variables to be solved. The equation solver identifies derived variables and eliminates them from the system of equations. The equation solver considers the remaining variables to be primitive variables. The primitive variables may be rewritten into a representation of the system of equations or into a set of equations that may be used to solve for values of the primitive variables. The equation solver solves for values of the primitive variables. Prior to solving or during solving iterations, the equation solver may apply storage policies to further reduce the number of variables to be solved. The storage policies indicate parameters and techniques for eliminating primitive variables to be solved, such as primitive variables that are insignificant (i.e., have little effect on a solution for the system of equations).