# A consistent analytical formulation for volume-estimation of geometries enclosed by implicitly defined surfaces

## TL;DR

This paper introduces a general analytical method for accurately estimating the volume of geometries enclosed by implicit surfaces, ensuring consistency across mesh refinements with demonstrated second-order accuracy.

## Contribution

It provides a unified analytical formulation for 2D and 3D volume estimation that addresses mesh refinement inconsistencies.

## Key findings

- Achieves 2nd-order accuracy in volume estimation
- Applicable to all 2D cases and elementary 3D cases
- Ensures consistency across mesh refinements

## Abstract

We have derived an analytical formulation for estimating the volume of geometries enclosed by implicitly defined surfaces. The novelty of this work is due to two aspects. First we provide a general analytical formulation for all two-dimensional cases, and for elementary three three-dimensional cases by which the volume of general three-dimensional cases can be computed. Second, our method addresses the inconsistency issue due to mesh refinement. It is demonstrated by several two-dimensional and three-dimensional cases that this analytical formulation exhibits 2nd-order accuracy.