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Shear strain localization refers to the phenomenon of accumulation of material deformation in narrow slip zones. Many materials exhibit strain localization under different spatial and temporal scales, particularly rocks, metals, soils, and concrete. In the Earth's crust, irreversible deformation can occur in brittle as well as in ductile regimes. Modeling of shear zones is essential in the geodynamic framework. Numerical modeling of strain localization remains challenging due to the non‐linearity and multi‐scale nature of the problem. We develop a numerical approach based on graphical processing units (GPU) to resolve the strain localization in two and three dimensions of a (visco)‐hypoelastic‐perfectly plastic medium. Our approach allows modeling both the compressible and incompressible visco‐elasto‐plastic flows. In contrast to symmetric shear bands frequently observed in the literature, we demonstrate that using sufficiently small strain or strain rate increments, a non‐symmetric strain localization pattern is resolved in two‐ and three‐dimensions, highlighting the importance of high spatial and temporal resolution. We show that elasto‐plastic and visco‐plastic models yield similar strain localization patterns for material properties relevant to applications in geodynamics. We achieve fast computations using three‐dimensional high‐resolution models involving more than 1.3 billion degrees of freedom. We propose a new physics‐based approach explaining spontaneous stress drops in a deforming medium. Plain Language Summary: Strain localization is the accumulation of strain in narrow regions of rocks and other materials like metals, soils, and concrete, occurring at different scales. The strength of most geomaterials, particularly rocks, is strongly pressure‐dependent, with strength increasing with increasing pressure. We developed efficient numerical algorithms using High‐Performance Computing (HPC) and graphical processing units (GPUs) to model strain localization in 2D and 3D for applications in geodynamics and earthquake physics. Unlike previous models, our method reveals non‐symmetrical patterns by using very small strain increments, highlighting the need for high‐detail modeling. We found that elasto‐plastic and visco‐plastic models show similar strain patterns for relevant materials. Our method also achieves fast, detailed computations with over 1.3 billion variables and offers a new explanation for sudden stress drops in deforming materials. Key Points: We resolve material instability during deformation resulting in a non‐symmetric pattern of strain localizationWe demonstrate the similarity in patterns of strain localization between frictional and time‐dependent plasticity modelsWe achieve fast numerical simulations in high‐resolution model setups in three dimensions involving more than 500 million degrees of freedom [ABSTRACT FROM AUTHOR]

Biot's equations describe the physics of hydromechanically coupled systems establishing the widely recognized theory of poroelasticity. This theory has a broad range of applications in Earth and biological sciences as well as in engineering. The numerical solution of Biot's equations is challenging because wave propagation and fluid pressure diffusion processes occur simultaneously but feature very different characteristic time scales. Analogous to geophysical data acquisition, high resolution and three dimensional numerical experiments lately redefined state of the art. Tackling high spatial and temporal resolution requires a high‐performance computing approach. We developed a multi‐ graphical processing units (GPU) numerical application to resolve the anisotropic elastodynamic Biot's equations that relies on a conservative numerical scheme to simulate, in a few seconds, wave fields for spatial domains involving more than 1.5 billion grid cells. We present a comprehensive dimensional analysis reducing the number of material parameters needed for the numerical experiments from ten to four. Furthermore, the dimensional analysis emphasizes the key material parameters governing the physics of wave propagation in poroelastic media. We perform a dispersion analysis as function of dimensionless parameters leading to simple and transparent dispersion relations. We then benchmark our numerical solution against an analytical plane wave solution. Finally, we present several numerical modeling experiments, including a three‐dimensional simulation of fluid injection into a poroelastic medium. We provide the Matlab, symbolic Maple, and GPU CUDA C routines to reproduce the main presented results. The high efficiency of our numerical implementation makes it readily usable to investigate three‐dimensional and high‐resolution scenarios of practical applications. Key Points: We present the dimensional analysis of Biot's equationsWe perform three dimensional numerical simulations of poroelastic wave propagationWe propose a multi‐graphical processing units implementation resolving over 1.5 billion grid cells in a few seconds with near ideal parallel efficiency [ABSTRACT FROM AUTHOR]