标题： 用于涉及地质力学裂缝互补方程的代数系统的内点方法 显示英文标题

标题： Interior point methods for an algebraic system involving complementarity equations for geomechanical fractures

摘要： 许多应用如微地震断层建模、裂缝储层建模以及断层连通性的解释/验证涉及在由裂纹扰动的背景介质中求解椭圆边界值问题，这些裂纹表现为一个或多个表面（带边界）。当背景介质可被视为均匀时，边界积分方程成为求解裂缝问题的首选数值方法。采用这种方法，问题被重新表述为在裂缝表面提出的完全非局部方程。边界积分的离散化导致所谓的边界元方法（BEM），由于所考虑的算子的完全非局部性，这会导致密集填充的矩阵。离散化过程之后，地质学家会面临一个难以数值求解的方程组。地质学家已经提出了许多经验算法来求解这个方程组。不幸的是，理论上没有任何一种算法保证收敛（特别是当断层（裂缝）相互交叉形成几何上高度不规则的结构时）。实际上，它们似乎都不够鲁棒或高效。我们研究了另一种方法，称为内点法，该方法可以确保收敛（即使断层过于接近）。内点法在各种领域中已证明其效率，尤其是在线性规划中。在这里，尽管我们没有优化问题，但可以借鉴内点法的思想来数值求解所考虑的方程组。获得的数值结果展示了计算效率和准确性，突显了所实现方法的鲁棒性和有效性。 显示英文摘要

摘要： Many applications like subseismic fault modeling, fractured reservoir modeling and interpretation/validation of fault connectivity involve the solution to an elliptic boundary value problem in a background medium perturbed by the presence of cracks that take the form of one or many pieces of surface (with boundary). When the background medium can be considered as homogeneous, boundary integral equations appear as a method of choice for the numerical solution to fractures problems. With such an approach, the problem is reformulated as a fully non-local equation posed at the surface of cracks. Discretization of boundary integral resulting in the so-called Boundary Element Method (BEM) leads to densely populated matrices due to the full non-locality of the operators under consideration. After the discretization process, geologists are faced with a system of equations that turns out difficult to solve numerically. Many empirical algorithms have been proposed by geologists to solve this system of equations. Unfortunately, none of them is guaranteed to converge in theory (in particular when faults (fractures) intersect each other forming a geometrically highly irregular structure). In practice, none of them appears to be either robust or efficient. We investigate another approach, referred to as interior point methods, for which convergence can be ensured (even if faults are too close). Interior point methods have proved their efficiency in a wide variety of domains, most notably for linear programming. Here, even though we do not have any optimization problem, we can adapt ideas from interior point methods for the numerical resolution of the system considered. The numerical results obtained demonstrate computational efficiency and accuracy, highlighting the robustness and effectiveness of the implemented methods.