# Asymptotic analysis of singularly perturbed boundary value problems

Collaborators: V. Maz’ya and A. Movchan

The response of a material can be greatly influenced by the presence of defects situated within the medium. Small defects and inclusions in materials can appear through their manufacturing processes for example, in mold casting procedures (see Fig. 1(a)). In some cases, the introduction of defects into a material can be deliberate (see Fig. 1(b)), such as in the design of composite materials whose behaviour can be optimised for different practical purposes by the choice and positioning of the defects. Fig. 1: (a) A micrograph of a sample of slip casted titanium (image from https://www.mtm.kuleuven.be). (b) Laser drilled microholes in a composite (image from https://www.oxfordlasers.com)

When the materials are exposed to various loads, impurities may interact with each other, leading to the material’s catastrophic failure: an unwanted scenario if the material is part of civil engineering assembly!

Therefore, it is extremely important to understand how defects affect the behaviour of the material. Modelling the static and dynamic response of materials with impurities is generally a challenging task. The analytical computation of an exact solution to problems of this type is in most cases not an option. Moreover, conventional numerical techniques are not well suited to these problems, especially when physical fields rapidly oscillate or there are stress concentrations. In this case, an asymptotic model can be a useful tool in indicating how such materials behave.

Here, we consider a possible route to addressing this question that utilises the so-called method of mesoscale asymptotic approximations.

For this method, mathematically, one considers a boundary value problem (BVP) representing a solid with small defects (see Fig. 2(a)) under some loading. The approach used to develop the approximation to the solution of the problem requires model fields associated with

• the solid without defects (in Fig. 2(b)) and
• an infinite medium containing a single small defect (in Fig. 2(c)).

An approximation to the solution of the original BVP is then constructed using appropriate linear combinations of these fields that ensure the conditions for the BVP are satisfied to a high accuracy. Fig 2: (a) A representative solid with a region of small impurities. The asymptotic formulae rely on fields related to the problem in (b) the solid (without impurities) and (c) individual impurities in an infinite medium.

Mesoscale asymptotic approximations.

These approximations can be used to model granular materials, where large numbers of small impurities that interact closely interact with each other (such as in Fig 1(a) above). The shapes of the small impurities can be arbitrary and no assumptions governing the defect positions within the material are necessary (for instance consider the configuration in Fig. 2(a)).

The approximations or asymptotic formulae are able to efficiently capture the interaction between the small defects and can compete with finite element algorithms, used by industrialists and scientists in the design of materials. An example of such a scenario is shown in Fig 3. The formulae are also applicable in the scenarios when finite element schemes fail (see Fig 4). Fig. 3: Heat conduction problem for a spherical solid containing a cluster. (a) A spherical solid containing a cluster of 27 small spherical inclusions of different sizes. The inclusions are positioned in a non-periodic fashion. (b) A zoom of the cluster inside the solid. Here, red inclusions represent thermally insulated inclusions. The other colours represent different materials inside each inclusion (which are also different from those in the ambient medium). In (c) and (d), we show the contour plots for the absolute value of the temperature gradient in the solid (in (a)) that is subjected to a heat source. The results are shown along the cut plane indicated in (b). Presented in (c) are the computations based on the finite element algorithm and (d) those based on the mesoscale asymptotic approach. One can see there is an excellent agreement. Further details are given in .

The approximations have been developed for several problems in mathematical physics including low frequency vibration problems , elasticity [3, 4], electromagnetism and hydrostatics .

References

 Nieves, M.J. (2017): Asymptotic analysis of solutions to transmission problems in solids with many inclusions, SIAM J. Appl. Math. 77 (4), 1417-1443.
 Maz’ya, V.G., Movchan, A.B., Nieves, M.J. (2017): Eigenvalue problem in a solid with many inclusions: asymptotic analysis, Multiscale Model. Simul., 15(2), 1003–1047.
 Maz’ya, V.G., Movchan, A.B., Nieves, M.J (2016): Meso-scale models and approximate solutions for solids containing clouds of voids, Multiscale Model. Simul. 14, no.1, 138-172.
 Maz’ya, V.G., Movchan, A.B., Nieves, M.J (2014): Meso-scale approximations for solutions of the Dirichlet problem in a perforated elastic body, J. Math. Sci. (N.Y.) 202, no. 2, 215-244.
 Maz’ya, V.G., Movchan, A.B., Nieves, M.J (2013): Green’s Kernels and Meso-scale Approximations in Perforated Domains, Lecture Notes in Math. 2077, Springer.