Towards rationalised computational expense for simulating fracture over multiple scales

The project focuses on the numerical simulation of the failure of complex, heterogeneous structures. The simulation of such physical phenomena is of particular interest to practitioners as it enables to limit the number of destructive tests required to design and assess structures, and, ultimately, to decrease the safety factors used in design.

In such heterogeneous media, the description of crack or damage initiation and propagation must be done at the scale of the inhomogeneities (e.g. aggregates in a concrete structure) in order for the results to be predictive. If one uses such a fine-scale material model to simulate structures at an engineering scale (e.g. an aircraft composite panel or a concrete beam), very large numerical problems need to be solved. In addition, there is a strong need for engineers to run their models numerous times, for different sets of the design parameters (e.g. loading conditions, geometry or material properties).

Tackling such parametric multiscale problems is prohibitively expensive when using brute force parallel computing. However, one can use the fact that solutions to parametric problems usually evolve in a relatively coarse space: solutions to nearby parameter sets are usually close in a certain sense. This idea is classically used in Model Order Reduction, which proposes to reduce the size of the initial problem by several order of magnitude by simply reusing the information generated when solving the initial problem for several different sets of parameters.

However, in the case of fracture, the information provided by the initial problem is most of the time insufficient to describe the behaviour of the system for arbitrary parameters. Crack paths, defects, and subsequent ultimate strengths are strongly influenced by an even slight variation in the parameter set. Fortunately, we showed in our previous research that this characteristic only affects a local region surrounding the structural defects, whilst the behaviour far from these regions is remains relatively unchanged for a wide range of parameter values.

The proposed project will make use of this observation in a generic way, by coupling Reduced Order Modeling and Domain Decomposition. The structure will be divided in smaller subcomponents, on which Reduced Order Modeling will be applied separately. The consequence will be that the computational efforts will be greatly decreased in the regions that are far away from the damaged zone. Within the process zone itself, the substructuring framework will allow us to automatically switch to classical direct solvers.

In this sense, the research aims at rationalising the computational costs associated to the simulation of parametrised multiscale fracture simulations, by concentrating the numerical effort where it is most required and with minimal intervention of the user.

The project has the potential to have impact on engineering researchers working in various area of expertise where modelling over multiple scale is involved. The proposed method is mainly based on Linear Algebra, and can be used without significant modifications to solve problems in fluid dynamics, electromagnetics and nano-mechanics (see letters of support).

The fundamental part of the work addresses a problem where there is an intrinsic lack of separation of scales. This is one of the most challenging research areas in computational engineering and physics and this project will open new doors toward a new family of solvers for cheap and reliable numerical simulations of such problems.

Developers, designers, manufacturers will, in the medium to long term, benefit from cheaper numerical methods for multiscale problems, and in particular for fracture mechanics. It will enable them to incorporate more details in their models thereby allowing more optimised products. Fracture is important in most areas of engineering and science and this project will impact various domains of industry such as manufacturing, aerospace, mechanical, environmental, civil and electronics engineering (see letters of support).

In order to ensure that the beneficiaries, both industrial and academic, do benefit from the research, a workshop will be organised at the end of the 13th month of the programme. This will increase the visibility of the research at the academic level but also involve key stakeholders (stated below) to ensure that the next phases of the research are relevant to their needs. At this workshop, we will invite the relevant members of the UK Association for Computational Mechanics in Engineering as well as industrialists and stakeholders with whom we have ongoing and meaningful relationships (BAE Systems, Defense systems and technology laboratory Dstl, Rolls-Royce, EADS, European Space Agency, Cenaero, inuTech, ZenCrack). This workshop will be inscribed within the Multiscale Fracture (MultiFRAC FP7 IRSES (International Research Staff Exchange Scheme) exchange programme linking Weimar (Prof. Rabczuk, Germany), Cardiff (UK), Northwestern University (Prof. Ted Belytschko and colleagues, USA), University of Witwaterstrand (Dr. Muthu, South Africa), Indian Institute of Science Bangalore (Dr. Mahapatra, India), which will ensure dissemination beyond the UK.

The numerical developments will be carried out in MATLAB so as to allow a ``race to prototype'' for fast publication and module release as an Open Source toolbox (BSD(new) license). When a sufficient level of maturity of the MATLAB code is achieved, production codes will be developed by inuTech GmbH to ensure wide uptake of the work by industry and academia. Negotiations between the legal teams at Cardiff University and inuTech on licensing, intellectual property and non-disclosure are ongoing.

Soitec SA state in their letter of support that they will support the project financially by contributing to a President PhD Scholarship at Cardiff. The negotiations between the two parties' legal teams are still ongoing. This will ensure that the technology is directly applicable to the needs of high-end manufacturing industry, which is one of the focus directions to redress the UK Economy.