Summary The finite element method (FEM) and the spectral boundary integral method (SBI) have both been widely used in the study of dynamic rupture simulations along a weak interface. In this paper, we present a hybrid method that combines FEM and SBI through the consistent exchange of displacement and traction boundary conditions, thereby benefiting from the flexibility of FEM in handling problems with nonlinearities or small‐scale heterogeneities and from the superior performance and accuracy of SBI. We validate the hybrid method using a benchmark problem from the Southern California Earthquake Center's dynamic rupture simulation validation exercises.We further demonstrate the capability and computational efficiency of the hybrid scheme for resolving off‐fault heterogeneities by studying a 2D in‐plane shear crack in two different settings: one where the crack is embedded in a high‐velocity zone and another where it is embedded in a low‐velocity zone. Finally, we discuss the potential of the hybrid method for addressing a wide range of problems in geophysics and engineering.

The Finite Difference (FD) and the Spectral Boundary Integral (SBI) methods have been used extensively to model spontaneously propagating shear cracks in a variety of engineering and geophysical applications. In this paper, we propose a new modeling approach, in which these two methods are combined through consistent exchange of boundary tractions and displacements. Benefiting from the flexibility of FD and the efficiency of spectral boundary integral methods, the proposed hybrid scheme will solve a wide range of problems in a computationally efficient way. We demonstrate the validity of the approach using two examples for dynamic rupture propagation: one in the presence of a low velocity layer and the other in which off-fault plasticity is permitted. We discuss possible potential uses of the hybrid scheme in earthquake cycle simulations as well as an exact absorbing boundary condition.

Periodic systems have attracted a lot of attention in science and engineering due to the existence of band gaps in their frequency spectra. Here, we study the exural wave propagation in beams that are periodically connected in parallel and investigate how the contrast in the material and cross-sectional properties may aect the band structure of these systems and their dispersion properties. Results suggest that by changing the mass and stiness ratios of the two beams, or by changing the inter-beam connection compliance, several band gaps may emerge and that the band gap width, the lowest band gap edge frequency, as well as the nature of attenuation within the gap may be tuned. Furthermore, by considering a hierarchical system of periodically-connected beam elements with dierent unit cell sizes, we show how the interplay between scales may aect the overall dispersion properties of the system by opening and closing band gaps at dierent frequencies. These ndings suggest that a modular design approach may lead to novel dispersion properties in beam structures. Finally, using a Frequency Response Function approach, we show that the aforementioned results hold in the limit of nite structures.

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The propagation of acoustic and elastic waves in time-varying, spatially homogeneous media can exhibit different phenomena when compared to traditional spatially varying, temporally homogeneous media. In the present work, the response of a one-dimensional phononic lattice with time-periodic elastic properties is studied with experimental, numerical and theoretical approaches in both linear and nonlinear regimes. The system consists of repelling magnetic masses with grounding stiffness controlled by electrical coils driven with electrical signals that vary periodically in time. For small-amplitude excitation, in agreement with linear theoretical predictions, wave-number band gaps emerge. The underlying instabilities associated to the wave-number band gaps are investigated with Floquet theory and the resulting parametric amplification is observed in both theory and experiments. In contrast to genuinely linear systems, large-amplitude responses are stabilized via the nonlinear nature of the magnetic interactions of the system, and results in a family of nonlinear time-periodic states. The bifurcation structure of the periodic states is studied. It is found the linear theory correctly predicts parameter values from which the time-periodic states bifurcate from the zero state. In the presence of an external drive, the parametric amplification induced by the wave-number band gap can lead to bounded and stable responses that are temporally quasiperiodic. Controlling the propagation of acoustic and elastic waves by balancing nonlinearity and external modulation offers a new dimension in the realization of advanced signal processing and telecommunication devices. For example, it could enable time-varying, cross-frequency operation, modeand frequency-conversion, and signal-to-noise ratio enhancements.