The Leibniz University Hannover, long form in German Gottfried Wilhelm Leibniz Universität Hannover, is a public research university located in Hanover, Germany. Founded on 2 May 1831, it is one of the largest and oldest science and technology universities in Germany. In the 2014/15 school year it enrolled 25,688 students, of which 2,121 were from foreign countries. It has nine faculties which offer 190 full and part degree programs in 38 fields of study. It was named University of Hannover in 1978. In 2006, it was named after Gottfried Wilhelm Leibniz, the 18th century mathematician and philosopher. In 2018, Leibniz University Hannover was adopted as the official English name.Leibniz University Hannover is a member of TU9, an association of the nine leading Institutes of Technology in Germany. It is also a member of the Conference of European Schools for Advanced Engineering Education and Research (CESAER), a non-profit association of leading engineering universities in Europe. The university sponsors the German National Library of Science and Technology (TIB), the largest science and technology library in the world.
Structural global reliability evaluation under dynamic excitations often involves highly complex failure boundaries and multiple interacting limit states, resulting in prohibitively high computational cost when using traditional probabilistic integration methods. This paper aims to address this challenge by proposing an enhanced computational framework based on the decoupled multi-dimensional probability density evolution method (M-PDEM) combined with an extended absorbing boundary condition strategy. Specifically, the proposed framework achieves both theoretical and numerical decoupling by transforming the original high-dimensional integration in the space of basic random variables into a series of one-dimensional generalized density evolution equations (GDEEs) defined in the response space. Rigorous theoretical conditions are established to guarantee decoupling feasibility, requiring separability of the residual joint probability density function (PDF) and factorization of the failure domain into finite unions or differences of hyper-rectangular sets. A robust three-step numerical implementation is derived, incorporating (i) discrepancy-optimized probabilistic space partitioning, (ii) finite-volume solution of the GDEEs using a weighted essentially non-oscillatory (FV-WENO) scheme, and (iii) rule-based aggregation of reliability metrics for AND-, OR-, and general Boolean combinations of correlated limit-state events. Numerical examples, including a single-degree-of-freedom (SDOF) oscillator with AND-type limit states, an Euler–Bernoulli beam with OR-type limit states failure, and a nonlinear concrete frame under seismic loads with a general limit-state specification, validate that the proposed methodology substantially reduces computational complexity while preserving high accuracy (with relative errors consistently below 1%) at a moderate computational cost, with the accuracy controlled by the number of representative points used in the probabilistic-space partition. Overall, this enhanced decoupled M-PDEM with absorbing boundaries provides a unified, efficient, and accurate solution to structural dynamic reliability problems involving multiple and potentially coupled failure criteria.
Fractured rock masses exhibit complex deformation and failure behavior governed by the interaction between stress redistribution and evolving internal structure. However, existing numerical approaches struggle to simultaneously represent diffuse damage within intact material and discrete fracture along structural interfaces. To address this limitation, this study proposes a coupled finite–discrete element method and phase-field method (FDEM–PFM) framework that integrates bulk damage evolution with interfacial decohesion and block interaction in a unified formulation. Hereby, the phase-field method describes mesh-independent crack initiation and diffuse damage in the bulk, while the finite-discrete element method captures interface opening, sliding, contact, and post-failure block kinematics. This coupling enables a physically consistent transition from continuum damage to discrete fracture, overcoming the limitations of conventional approaches that treat these mechanisms separately. The predictive capability of the model is first assessed through validation against three loading configurations: uniaxial compression, direct shear, and biaxial compression under varying confinement. The framework accurately reproduces stress–strain responses, failure patterns, and pressure-dependent strength, demonstrating its ability to capture bulk damage, shear-dominated interface failure, and confinement effects within a single modeling approach. Building on this validation, a parametric study is conducted to investigate the influence of interfacial stiffness on stress transfer, energy partitioning, and failure mode transition. The results reveal three distinct regimes. At high stiffness, deformation is governed by bulk damage in a stress-controlled regime. At intermediate stiffness, bulk damage and interface failure interact, producing a coupled structure–stress response. At low stiffness, deformation localizes near interfaces, leading to a strain-controlled regime dominated by phase-field damage. These findings demonstrate that interfacial stiffness governs the transition between fundamentally different failure mechanisms and plays a central role in the macroscopic response of fractured rock. The proposed FDEM–PFM framework provides a robust and physically consistent tool for modeling structure–stress interactions, bridging the gap between continuum damage mechanics and discrete fracture dynamics.
This work presents a thermodynamically consistent model for flexoelectric fracture analysis utilizing truncated hierarchical B-splines (THB-splines) and an adaptive phase-field method. The model integrates elastic, gradient elastic, piezoelectric and flexoelectric effects, solving the problem as a coupled system, while crack propagation is addressed within a phase-field formulation. By developing a novel adaptive THB discretization strategy, the requirements of both C1 continuity and local mesh refinement are automatically fulfilled. To overcome the challenge of reproducing the flexoelectric crack-tip behavior, three different degradation functions are assigned to the elastic, gradient elastic and dielectric energies. The model is validated through representative numerical examples, providing insight into the interaction mechanisms between flexoelectricity and fracture. Fracture gives rise to a giant flexoelectric tip effect while simultaneously restricting the peak electrical response. In turn, flexoelectricity not only strengthens the fracture resistance of materials but also alters crack paths through the application of an electric field. The study shows that the presence of cracks substantially enhances flexoelectric sensitivity as well as energy conversion efficiency, providing theoretical support for the development of novel sensing and energy harvesting devices.
Third medium contact provides a smooth continuum alternative to classical contact algorithms by replacing explicit contact constraints with a highly compliant fictitious medium. In this work, an auxiliary-field stabilization is introduced in which a deformation-gradient-like field is treated as an independent unknown in the third medium and coupled to the physical deformation gradient by a penalty term. A gradient contribution acting on the auxiliary field provides the regularization mechanism without requiring a direct evaluation of higher displacement derivatives. Linear and quadratic interpolation spaces are investigated, including continuous and element-wise discontinuous auxiliary-field approximations. The numerical results show that continuous low-order auxiliary fields provide an effective gradient-type stabilization of the third medium, even when the displacement field is approximated by first-order finite elements. For element-wise discontinuous auxiliary fields, the additional unknowns remain local to each element and can be eliminated locally by static condensation, so that the global system does not necessarily contain additional auxiliary degrees of freedom. Benchmark problems involving large deformation, progressive self-contact and severe third-medium compression are used to assess the formulation.
The equal-area criterion is a graphical method for analyzing transient stability that has been known since the 1920s. It is intuitive, easy to understand, and used primarily in academic education. However, the integral of power P over transient angle δ′ used to calculate the acceleration area Facc and deceleration area Fdec in the P-δ′-diagram has no physical significance, since it describes power but not energy. This leads to the further question of what these areas actually represent. In this paper the energy changes during rotor acceleration and deceleration and their relationships to the areas Facc and Fdec are analyzed and obvious contradictions between assumptions and used swing equation, such as the usual assumption of constant turbine power, are explained. Key findings are that the areas Facc and Fdec only represent a small portion of the energy exchanged during the acceleration or deceleration phase, and that a M-δ ′-representation (M≙torque) should be used instead of the usual P-δ ′-diagrams in order to remain physically correct.
