Periodic Reporting for period 1 - MATT (Multiscale Analysis of Phase Transformations in Thermoelasticity)
Reporting period: 2022-09-01 to 2024-08-31
The overarching goal of the MATT project (Multiscale Analysis of Phase Transformations in Thermoelasticity) is to rigorously derive, analyze, and simulate efficient and accurate mathematical models for complex multiscale processes. These processes, which require careful consideration of evolving small-scale internal surfaces over time and space, are called multiscale moving boundary problems or MultMBPs. A prime example is the modeling of mechanical microstructural changes in steel under rapid temperature changes (Bainite formation from Austenite). Other real-world applications leading to the same class of mathematical problems include swelling of porous media, growth of tumors, and thawing of glaciers/permafrost - issues increasingly relevant in engineering and environmental contexts. In a world, where the need for specialized, dynamic, and complex materials (e.g. composites, meta-materials, bio-engineered materials) is ever increasing, increasing the mathematical understanding of their effective material behavior is particularly relevant.
The scale heterogeneity, introduced by numerous small, evolving internal boundaries, renders numerical simulations infeasible. For that reason, one must identify simplified models that are able to accurately describe and predict the material behavior while still being simple enough to allow for fast numerical simulations within the expected physical range. The mathematical derivation of such models is made complicated by the inherent non-linear structure of MultMBPs and requires new results regarding uniform estimates and compactness arguments. But even after a homogenization limit procedure, the limit models still suffer from complex scale-interactions making numerical simulations computationally expensive. As a consequence, smart and efficient numerical schemes are needed to tackle these limit model. Here, the project proposes a novel precomputing approach where certain calculations are shifted and parallelized into an offline phase. With this approach, simulations can be done much faster without sacrificing much accuracy.
The scale heterogeneity, introduced by numerous small, evolving internal boundaries, renders numerical simulations infeasible. For that reason, one must identify simplified models that are able to accurately describe and predict the material behavior while still being simple enough to allow for fast numerical simulations within the expected physical range. The mathematical derivation of such models is made complicated by the inherent non-linear structure of MultMBPs and requires new results regarding uniform estimates and compactness arguments. But even after a homogenization limit procedure, the limit models still suffer from complex scale-interactions making numerical simulations computationally expensive. As a consequence, smart and efficient numerical schemes are needed to tackle these limit model. Here, the project proposes a novel precomputing approach where certain calculations are shifted and parallelized into an offline phase. With this approach, simulations can be done much faster without sacrificing much accuracy.
The overall work was conducted via five work packages, three scientific work packages (WP1: Mathematical analysis of the ε-problem, WP2: Limit analysis of the limit problem, and WP3: Numerical simulations, numerical analysis, and model validation), one work package dedication to dissemination and communication, and finally a work package concerned with training activities.
Regarding the scientific work and output, it has led to three peer-reviewed publications and several preprints which are currently under review. In addition, eight invited talks were given at conferences & workshops all over the world and two small workshops and one mini-symposium were organized in Karlstad during the action.
WP1: Starting with a simplified protoype MultMBP (one-phase Stefan type thermo-elasticity problem with locally averaged normal velocities describing the phase evolution), well-posedness and uniform estimates were established via the Hanzawa transformation. Crucially, it was shown that the existence interval for the solutions is independent of the scale parameter. Based on these results further models (two-phase systems, swelling/clogging of porous media) were studied.
WP2: A rigorous limit analysis for the prototype problem was conducted in the context of two-scale convergence based on the results from WP1. In addition, the limit problem was comprehensively studied regarding uniqueness (available under certain regularity assumptions) and long-time behaviour (available under certain smallness assumptions). Certain error estimates necessary for the numerical treatment in WP3 were also investigated. The follow-up work is ongoing of extending these results to other problems is ongoing, with partial results achieved for two-phase systems and clogging of porous media.
WP3: A precomputing scheme has been introduced to efficiently an accurately deal with the intricately and nonlinearly coupled scale-interactions of the limit problems. This scheme, where part of the expensive calculations of effective parameters is pushed into an offline phase and later interpolated, has been shown to be stable and significantly faster. Also, the additional error is controllable. Moreover, the limit models were validated by direct numerical comparison with the initial heterogeneous models.
Regarding the scientific work and output, it has led to three peer-reviewed publications and several preprints which are currently under review. In addition, eight invited talks were given at conferences & workshops all over the world and two small workshops and one mini-symposium were organized in Karlstad during the action.
WP1: Starting with a simplified protoype MultMBP (one-phase Stefan type thermo-elasticity problem with locally averaged normal velocities describing the phase evolution), well-posedness and uniform estimates were established via the Hanzawa transformation. Crucially, it was shown that the existence interval for the solutions is independent of the scale parameter. Based on these results further models (two-phase systems, swelling/clogging of porous media) were studied.
WP2: A rigorous limit analysis for the prototype problem was conducted in the context of two-scale convergence based on the results from WP1. In addition, the limit problem was comprehensively studied regarding uniqueness (available under certain regularity assumptions) and long-time behaviour (available under certain smallness assumptions). Certain error estimates necessary for the numerical treatment in WP3 were also investigated. The follow-up work is ongoing of extending these results to other problems is ongoing, with partial results achieved for two-phase systems and clogging of porous media.
WP3: A precomputing scheme has been introduced to efficiently an accurately deal with the intricately and nonlinearly coupled scale-interactions of the limit problems. This scheme, where part of the expensive calculations of effective parameters is pushed into an offline phase and later interpolated, has been shown to be stable and significantly faster. Also, the additional error is controllable. Moreover, the limit models were validated by direct numerical comparison with the initial heterogeneous models.
There are two main avenues where MATT have pushed beyond the state of the art: (i) rigorously deriving effective models for MultMBPs and (ii) introducing and analyzing a precomputing scheme able to efficiently tackle scale-coupled limit problems.
(i) The well-known Hanzawa method was used in conjunction with fixed-point arguments to show well-posedness and uniform estimates for MultMBPs with numerous small, evolving internal surfaces. Based on these results, limit models were rigorously derived, which before was only possible by either prescribing the evolution or by using phase-field models.
(ii) The limit models are still rather complicated to simulate due to complex scale interactions (the evolving, microscopic surfaces influences the macroscopic material properties). A precomputing approach was introduced and investigated, where the calculations of possible evolutions was pushed into an offline phase, speeding up the simulations significantly. It was also shown that the error introduced by the resulting interpolation can be sufficiently controlled.
Especially the precomputing approach has a huge expected potential impact considering the fact that it is not restricted to MultMBPs problems. Indeed, the same approach can be used in a wide class of multiscale problems with complex, non-linear interactions. The same idea was already used in a model describing the nonlinear dispersion of fluids in porous media.
(i) The well-known Hanzawa method was used in conjunction with fixed-point arguments to show well-posedness and uniform estimates for MultMBPs with numerous small, evolving internal surfaces. Based on these results, limit models were rigorously derived, which before was only possible by either prescribing the evolution or by using phase-field models.
(ii) The limit models are still rather complicated to simulate due to complex scale interactions (the evolving, microscopic surfaces influences the macroscopic material properties). A precomputing approach was introduced and investigated, where the calculations of possible evolutions was pushed into an offline phase, speeding up the simulations significantly. It was also shown that the error introduced by the resulting interpolation can be sufficiently controlled.
Especially the precomputing approach has a huge expected potential impact considering the fact that it is not restricted to MultMBPs problems. Indeed, the same approach can be used in a wide class of multiscale problems with complex, non-linear interactions. The same idea was already used in a model describing the nonlinear dispersion of fluids in porous media.