Cartesian Grid Contact Solving via Level-Set Triangulation
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Solution Overview
Problem
The finite element method struggles to effectively model contact between deformable solids on a Cartesian grid due to geometrical non-conformity and nonlinear boundary conditions, which complicates numerical solutions and requires significant computational resources.
Innovation Solution
A level-set description of geometry is used to triangularize boundaries on a Cartesian grid, enabling surfacic Gauss integration and contact integral computation, which are then interpolated onto a regular grid for solving contact problems without adapting meshes, utilizing a fictitious domain framework and surface-to-surface contact method.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If a Cartesian grid discretization is used for finite element modeling, then computational efficiency and resource utilization are improved, but the ability to accurately represent geometric boundaries and contact surfaces deteriorates
Solution Approach 1:
The boundary representation is segmented into two components: a coarse Cartesian grid for volumetric discretization and a fine level-set triangulation for surface representation. This segmentation allows each component to excel at its appropriate scale - the Cartesian grid provides computational efficiency for bulk material modeling while the level-set triangulation delivers geometric precision for contact surfaces.
Solution Approach 2:
The patent transitions from a purely 3D volumetric Cartesian grid to a hybrid representation that incorporates a 2D level-set function defined on the grid nodes. This level-set function implicitly defines the boundary surface, adding a dimensional layer of geometric control without disrupting the underlying 3D Cartesian structure.
2Manufacturing precision
If mesh adaptation is performed to conform to geometry, then contact surface accuracy is improved, but computational cost and complexity increase significantly
Solution Approach 1:
The level-set function is constructed beforehand to implicitly capture the geometric boundary and contact surfaces. This preliminary action of defining geometry through the level-set function eliminates the need for subsequent mesh adaptation operations, as the Cartesian grid remains fixed while the level-set provides accurate surface representation.
Solution Approach 2:
The level-set function serves as an intermediary between the simple Cartesian grid and the complex contact geometry. Instead of adapting the grid to fit the geometry, the level-set mediates by providing a mathematical representation of the boundary that can be queried during contact detection and force calculation.
3Reliability
If nonlinear contact boundary conditions are enforced on non-conforming meshes, then contact constraint accuracy deteriorates, but solving complexity increases
Solution Approach 1:
The patent replaces the traditional mechanical approach of enforcing contact constraints through nodal projections with a surface integral formulation. Instead of projecting contact forces onto discrete nodes, the method integrates contact pressures over the continuous contact surface defined by the level-set, providing more accurate stress distribution without requiring complex constraint algorithms.
Data Source
AI summary
A contact problem for characterizing physical contact between two or more bodies is solved by receiving a level-set description of a geometry in a Cartesian grid comprising a plurality of cells. The geometry includes a boundary. Thereafter, the boundary is triangularized for each cell intersecting the boundary to result in a plurality of triangles. A surfacic Gauss integration rule is generated on each triangle using an interpolation of volumetric coordinates. Subsequently, a target point is computed for each Gauss point that corresponds to a projection of such Gauss point on a target surface. The computed target points are then used to integrate contact on the triangles. The problem can then be solved to compute a displacement field respecting contact conditions. Related apparatus, systems, techniques and articles are also described.