A three-scale domain decomposition method for the 3D analysis of debonding in laminates

In electronic devices solder joints form a mechanical as well as an electrical connection between the circuit board and the component (e.g. a chip or a resistor). Temperature variations occurring during field use cause crack initiation and crack growth inside the joints. Accurate prediction of the lifetime requires a method to simulate the damage process based on microstructural properties.Numerical simulation of developing cracks and microstructural entities such as grain boundaries and grain junctions gives rise to several problems. The solution contains strong and weak discontinuities as well as weak singularities. To obtain reasonable solutions with the finite element method (FEM) the element edges have to align with the cracks and the grain boundaries, which imposes geometrical restrictions on the mesh choice. Additionally, a large number of elements has to be used in the vicinity of the singularities which increases the computational effort. Both problems can be circumvented with the extended finite element method (X-FEM) by using appropriate enrichment functions.In this thesis the X-FEM will be developed for the simulation of complex microstructural geometries. Due to the anisotropy of the different grains forming a joint and the variety of different microstructural configurations it is not always possible to write the enrichment functions in a closed form. A procedure to determine enrichment functions numerically is explained and tested. As a result, a very simple meshing scheme, which will be introduced here, can be used to simulate developing cracks in solder joint microstructures. Due to the simplicity of the meshing algorithm the simulation can be automated completely. A large number of enrichment functions must be used to realize this. Well-conditioned equation systems, however, cannot be guaranteed for such an approach. To improve the condition number of the X-FEM stiffness matrix and thus the robustness of the solution process a preconditioning technique is derived and applied.This approach makes it possible to develop a new and fully automated procedure for addressing the reliability of solder joints numerically. The procedure relies on the random generation of microstructures. Performing crack growth calculations for a series of these structures makes it possible to address the influence of varying microstructures on the damage process. Material parameters describing the microstructure are determined in an inverse procedure. It will be shown that the numerical results correspond well with experimental observations.

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“…[31,32]). In this thesis domain decompositions will be used to develop a robust preconditioner for the X-FEM.…”

Section: Domain Decompositionmentioning

confidence: 99%

A robust preconditioning technique for the extended finite element method

Menk

Bordas

2010

Numerical Meth Engineering

140

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“…In the latter, the traction-separation law for a macro crack is determined on the fly, in the spirit of the FE 2 method [3], from the responses in a propagating damaged zone at the microscale. Potential application of the proposed averaging scheme can be found in methods with nested scales computations [19,20,17] and with domain decomposition methods [28,29].…”

Section: Introductionmentioning

confidence: 99%

On the existence of representative volumes for softening quasi-brittle materials – A failure zone averaging scheme

Nguyen¹,

Lloberas-Valls²,

Stroeven³

et al. 2010

Computer Methods in Applied Mechanics and Engineering

133

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a b s t r a c tThe concept of the representative volume element (RVE) for softening materials is revised in this contribution. It is demonstrated by means of numerical simulations that there exists a sample which is statistically representative for quasi-brittle materials with random microstructure like concrete. This finding is an important ingredient for homogenization-based multiscale modelling of softening materials.

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“…Dedicated strategies have been developed to handle these very different scales that are required to simulate cracked bodies. These methods can use energy coupling methods [3,13], domain decomposition methods [12,16], homogenization [8,10,1] or generalized FEM [17]. Among them a multiscale method has been recently proposed and adapted to X-FEM [27] which is based on a multigrid solver [5,25].…”

Section: Introductionmentioning

confidence: 99%

Direct estimation of generalized stress intensity factors using a three‐scale concurrent multigrid X‐FEM

Passieux

Gravouil

Réthoré

et al. 2010

Numerical Meth Engineering

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mation of generalized stress intensity factors using a three-scale concurrent multigrid X-FEM. International Journal for Numerical Methods in Engineering, Wiley, 2011, 85 (13) Abstract A concurrent multigrid method is devised for the direct estimation of stress intensity factors (SIF) and higher order coefficients of the elastic crack tip asymptotic field. The proposed method bridges three characteristic length scales that can be present in fracture mechanics: the structure, the crack and the singularity at the crack tip. For each of them, a relevant model is proposed. First, a truncated analytical reduced order model based on the Williams' expansion is used to describe the singularity at the tip. Then, it is coupled with a standard X-FEM model which is known to be suitable for the scale of the crack. A multigrid solver finally bridges the scale of the crack to that of the structure for which a standard FE model is often accurate enough. Dedicated coupling algorithms are presented and the effects of their parameters are discussed. The efficiency and accuracy of this new approach is exemplifyed using three benchmarks.

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A three-scale domain decomposition method for the 3D analysis of debonding in laminates

Cited by 72 publications

References 31 publications

A robust preconditioning technique for the extended finite element method

A robust preconditioning technique for the extended finite element method

On the existence of representative volumes for softening quasi-brittle materials – A failure zone averaging scheme

Direct estimation of generalized stress intensity factors using a three‐scale concurrent multigrid X‐FEM

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