Abstract
When using many modern methods of automatic generation of surface meshes of implicitly defined geometric objects, the accuracy of approximation in the vicinity of surface singularities (holes, breaks, etc.) is lost. To improve surface meshes of geometric objects, various methods of smoothing are used. The existing smoothing methods are focused on triangular elements, but optimization of surface meshes of geometric objects on the basis of elements of another shape (for example, quadrangles) is less studied. The paper proposes the mathematical apparatus based on the use of the energy functional for each model node. The proposed functional considers the distance from the current node to the adjacent nodes and the distance from the geometric centers of the incident elements to the surface. The algorithm for minimizing the energy functional for smoothing surface meshes of implicitly defined geometric objects is developed. The developed algorithm is a modification of the Gaussian method for the case of search for a minimum in the local coordinates of a polygon formed by neighboring elements. The algorithm is local: minimization is performed consistently for each model node, so its repeated application provides models with more accurate approximation of the boundary. The developed algorithm for minimizing the functional does not require the insertion of new nodes. As a consequence, it is possible, using a single procedure, to optimize meshes based on triangles, quadrangles or mixed type (containing triangles and quadrangles simultaneously). As a result, the accuracy of the approximation of surfaces in the vicinity of their singularities increases, as demonstrated by the examples of smoothing models of complex objects.
Highlights
Modeling of many natural and technical objects is associated with the need to describe their shape in the form of mathematical relations
A very common way of determining the set of points belonging to some object is to use implicit mathematical functions
Moreiterations, the surface break near the hole is over, methods of improving meshes based on topological approximated by the nodes and edges of the changes can additionelements, but the elements close in shape to ally be used in each iteration
Summary
Modeling of many natural and technical objects is associated with the need to describe their shape in the form of mathematical relations. A very common way of determining the set of points belonging to some object is to use implicit mathematical functions. Implicit functions can be constructed step-by-step using set-theoretic operations (negation, disjunction, conjunction). Such operations are implemented in the form of systems of R-functions [1, 2]. In the problems of computer graphics and numerical analysis, it becomes necessary to construct meshes of the boundaries of the objects under study. The use of existing methods [3,4,5] for generating meshes of the boundaries of implicit functions can lead to an inaccurate approximation of the boundary near features (holes, breaks, etc.). It is important to develop methods for improving meshes of the boundaries of geometric objects that do not require an increase in the number of nodes
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