The mathematical algorithm of multi-point constraints in the simulations of three-dimensional Numerical Manifold Method

In this paper, we introduce the parameter adjustment method with condition equations(one surveying adjustment method in geodetic data processing) to three-dimensional Manifold Method through formula derivation, and present the strict-constraint solution and least-squares solution strategies. In least-square solution, we develop the power conception of surveying adjustment and use power ratio to balance the physical and mathematical equations. Then, we use the uniaxial tensile model to verify the validity of above two solution strategies, and analyze their difference. Furthermore, the shearing failure simulation with mathematical constraint is presented. In conclusion, the essential difference of above two strategies is that the strict-constraint strategy can realize strong constraint on some unknowns and have minimum influence on others in the examples of this paper. On the other hand, the least-square strategy influences more than constrained unknowns, and perhaps affects the whole equations. Furthermore, we can control the constraint intensity by adjusting power ratio when using least-square strategy, because the constraint intensity is directly proportional to the power ratio.