Inverse Algorithm for Weierstrass-Mandelbrot Surface Characterization

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Solution Overview

Problem

Current methods for determining the parameters of a Weierstrass-Mandelbrot analytical representation of rough surfaces are limited, as they primarily focus on fractal dimension and enforce restrictive assumptions, failing to accurately identify all critical parameters necessary for comprehensive surface characterization.

Innovation Solution

A computer-implemented method is developed to directly determine all parameters of the Weierstrass-Mandelbrot function, including phases and frequency density, using an inverse algorithm applied to elevation data, enabling the identification of mechanical, thermal, and electrical properties of sliding contacts and surface topographies.

Engineering Contradictions & Design Principles

VSEngineering Contradiction Analysis

1Adaptability or versatility

If the power spectrum method is used to determine fractal dimension, then the method is widely applicable, but it can only identify one parameter (fractal dimension) and enforces restrictive assumptions about other parameters

Engineering Contradiction:

Improvewide applicabilityVSAvoidparameter identification accuracy

Core Design Contradiction:

Adaptability or versatilityVSMeasurement precision

Solution Approach 1:

The patent inverts the traditional approach by using an inverse algorithm that determines all W-M parameters (including fractal dimension, amplitude, and frequency density) simultaneously from surface elevation data, rather than using the power spectrum method that only extracts fractal dimension while assuming other parameters. This inversion resolves the contradiction by achieving both wide applicability and complete parameter identification accuracy.

Inventive Principle:

Principle #13The other way round (Inversion)

Solution Approach 2:

The patent changes the parameter determination approach from indirect estimation (power spectrum method that assumes parameter values) to direct determination through inverse algorithms. By formulating and solving an inverse problem that directly computes all W-M parameters from measured surface data, the method eliminates restrictive assumptions and achieves precise identification of all critical parameters including fractal dimension, amplitude, and frequency density.

Inventive Principle:

Principle #35Parameter changes

2Device complexity

If only fractal dimension is determined, then the method is simpler, but it fails to identify all critical parameters necessary for comprehensive surface characterization

Engineering Contradiction:

Improvemethod simplicityVSAvoidsurface characterization completeness

Core Design Contradiction:

Device complexityVSLoss of information

Solution Approach 1:

The patent creates a universal inverse algorithm that simultaneously determines all W-M parameters (fractal dimension, amplitude, frequency density, and phases) from surface elevation data. This multi-functional approach resolves the contradiction by achieving comprehensive surface characterization without requiring separate methods for each parameter, thus maintaining relative simplicity while eliminating information loss.

Inventive Principle:

Principle #6Universality (Multi-functionality)

Solution Approach 2:

The patent segments the parameter determination process into distinct computational steps within the inverse algorithm framework: forming the system of equations from elevation data, applying regularization techniques, and sequentially solving for different parameter sets. This segmentation allows the complex multi-parameter determination to be achieved through a structured, manageable process that maintains methodological simplicity while achieving complete characterization.

Inventive Principle:

Principle #1Segmentation

3Ease of manufacture

If restrictive assumptions are enforced about parameter values, then the computational process is simplified, but the accuracy of surface representation deteriorates

Engineering Contradiction:

Improvecomputational simplicityVSAvoidsurface representation accuracy

Core Design Contradiction:

Ease of manufactureVSManufacturing precision

Solution Approach 1:

The patent applies dynamic regularization techniques that adaptively adjust computational constraints during the inverse algorithm execution. Rather than enforcing fixed restrictive assumptions, the method dynamically determines appropriate parameter ranges and constraints based on the specific surface data being analyzed, resolving the contradiction by maintaining computational tractability while achieving accurate surface representation through adaptive, data-driven parameter selection.

Inventive Principle:

Principle #15Dynamics

Data Source

PatentUS8884954B2Algorithm and a method for characterizing surfaces with fractal nature

Publication Date: 2014.11.11 THE GOVERNMENT OF THE UNITED STATES OF AMERICA AS REPRESENTED BY THE SECRETARY DEPARTMENT OF HEALTH & HUMAN SERVICES

AI summary

A computer implemented method for directly determining parameters defining a Weierstrass-Mandelbrot (W-M) analytical representation of a rough surface scalar field with fractal character, embedded in a three dimensional space, utilizing pre-existing measured elevation data of a rough surface in the form of a discrete collection of data describing a scalar field at distinct spatial coordinates, is carried out by applying an inverse algorithm to the elevation data to thereby determine the parameters that define the analytical and continuous W-M representation of the rough surface. The invention provides a comprehensive approach for identifying all parameters of the W-M function including the phases and the density of the frequencies that must greater than 1. This enables the infinite-resolution analytical representation of any surface or density array through the W-M fractal function.