Guillermo Rus

Optimization algorithms for identification inverse problems with the boundary element method

In this paper the most suitable algorithms for unconstrained optimization now available applied to an identification inverse problem in elasticity using the boundary element method (BEM) are compared. Advantage is taken of the analytical derivative of the whole integral equation of the BEM with respect to the variation of the geometry, direct differentiation, which can be used to obtain the gradient of the cost function to be optimized.

A two-parameter model of the effective elastic tensor for cortical bone.

Multiscale models of cortical bone elasticity require a large number of parameters to describe the organization and composition of the tissue. We hypothesize that the macro-scale anisotropic elastic properties of different bones can be modeled retaining only two variable parameters, and setting the others to universal values identical for all bones. Cortical bone is regarded as a two-phase composite material: a dense mineralized matrix (ultrastructure) and a soft phase (pores). The ultrastructure is assumed to be a homogeneous and transversely isotropic tissue whose elastic properties in different directions are mutually dependent and can be scaled with a single parameter driving the overall rigidity. This parameter is taken to be the volume fraction of mineral f(ha). The pore network is modeled as an ensemble of water-filled cylinders and described only by the porosity p. The effective macroscopic elasticity tensor C(ij)(f(ha),p) is calculated with a multiscale micromechanics approach starting from existing models. The modeled stiffness coefficients compare favorably to four literature datasets which were chosen because they provide the full stiffness tensors of groups of human samples. Since the physical counterparts of f(ha) and p were unknown for the datasets, their values which allow the best fit of experimental tensors by the modeled ones were determined by optimization. Optimum values of f(ha) and p are found to be unique and realistic. These results suggest that a two-parameter model may be sufficient to model the elasticity of different samples of human femora and tibiae. Such a model would in particular be useful in large-scale parametric studies of bone mechanical response.

Approximate Bayesian computation by subset simulation

A new approximate Bayesian computation (ABC) algorithm for Bayesian updating of model parameters is proposed in this paper, which combines the ABC principles with the technique of subset simulation for efficient rare-event simulation, first developed in S. K. Au and J. L. Beck [Probabilistic Engrg. Mech., 16 (2001), pp. 263-277]. It has been named ABC-SubSim. The idea is to choose the nested decreasing sequence of regions in subset simulation as the regions that correspond to increasingly closer approximations of the actual data vector in observation space. The efficiency of the algorithm is demonstrated in two examples that illustrate some of the challenges faced in real-world applications of ABC. We show that the proposed algorithm outperforms other recent sequential ABC algorithms in terms of computational efficiency while achieving the same, or better, measure of accuracy in the posterior distribution. We also show that ABC-SubSim readily provides an estimate of the evidence (marginal likelihood) for posterior model class assessment, as a by-product.

Robust parametrization for non-destructive evaluation of composites using ultrasonic signals

Anticipating and characterizing damages in layered carbon fiber-reinforced polymers is a challenging problem. Non-destructive evaluation using ultrasonic signals is a well-established method to obtain physically relevant parameters to characterize damages in isotropic homogeneous materials. However, ultrasonic signals obtained from composites require special care in signal interpretation due to their structural complexity. In this paper, some enhancements on the interpretation are done by adapting classical parametrization techniques to extract relevant features from the ultrasonic signals. Thus, a cepstral-based feature extractor is firstly designed and optimized by using a classification system based on cepstral distances. Then, this feature extractor is applied in an analysis-by-synthesis scheme which, by using a numerical model of the specimen, infers the values of the damage parameters.

Analysis and design of wedge transducers using the boundary element method

Cones or wedges inserted between an ultrasonic transducer and a specimen enhances certain characteristics of the transducers. Such an arrangement is useful in that the transducer can be used for transmitting and receiving signals on a point (or line) source, which can eliminate the undesirable aperture effect that makes the transducer blind to waves traveling in certain directions and to those of certain frequencies. In this paper, a comprehensive numerical analysis based on a wave propagation model is carried out to study the characteristics and parameters of wedges. We study the effect of dimensions, shape and aperture on frequency response and directivity. For computational accuracy and efficiency, the boundary element method is used in the analysis.

Mechanical assessment of cervical remodelling in pregnancy: insight from a synthetic model.

During the gestation and the cervical remodelling, several changes occur progressively in the structure of the tissue. An increase in the hydration, disorganisation of collagen network and decrease in elasticity can be observed. The collagen structure disorganisation is particularly complex: collagen fibres turn thicker and more wavy as the gestation progresses in a transition from relatively straight fibres to wavy fibres, while pores between collagen fibres become larger and separated. Shear wave elastography is a promising but not yet fully understood tool to assess these structural changes and the cervix׳s ability to dilate. To this end, a numerical histo-mechanical model is proposed in the present study, which aims at linking variations in the microscopic histo-biomechanical processes with shear wave propagation characteristics. Parametric simulations are carried out for a broad range of mechanical and geometrical parameters. Results show a direct relationship between the histological and morphological changes during pregnancy and the viscoelastic behaviour of the tissue.

Probabilistic inverse problem to characterize tissue-equivalent material mechanical properties

The understanding of internal processes that affect the changes of consistency of soft tissue is a challenging problem. An ultrasound-monitoring Petri dish has been designed to monitor the evolution of relevant mechanical parameters during engineered tissue formation processes in real time. A better understanding of the measured ultrasonic signals required the use of numerical models of the ultrasound-tissue interactions. The extraction of relevant data and its evolution with sufficient sensitivity and accuracy is addressed by applying well-known signal processing techniques to both the experimental and numerically predicted measurements. In addition, a stochastic model-class selection formulation is used to rank which of the proposed interaction models are more plausible. The sensitivity of the system is verified by monitoring a gelation process.

Logical inference for inverse problems

Estimating a deterministic single value for model parameters when reconstructing the system response has a limited meaning if one considers that the model used to predict its behaviour is just an idealization of reality, and furthermore, the existence of measurements errors. To provide a reliable answer, probabilistic instead of deterministic values should be provided, which carry information about the degree of uncertainty or plausibility of those model parameters providing one or more observations of the system response. This is widely-known as the Bayesian inverse problem, which has been covered in the literature from different perspectives, depending on the interpretation or the meaning assigned to the probability. In this paper, we revise two main approaches: the one that uses probability as logic, and an alternative one that interprets it as information content. The contribution of this paper is to provide an unifying formulation from which both approaches stem as interpretations, and which is more general in the sense of requiring fewer axioms, at the time the formulation and computation is simplified by dropping some constants. An extension to the problem of model class selection is derived, which is particularly simple under the proposed framework. A numerical example is finally given to illustrate the utility and effectiveness of the method.

Assessing viscoelasticity of shear wave propagation in cervical tissue by multiscale computational simulation.

The viscoelastic properties are recently being reported to be particularly sensitive to the gestation process, and to be intimately related to the microstructure of the cervical tissue. However, this link is not fully understood yet. In this work, we explore the importance of the heterogeneous multi-scale nature of cervical tissue for quantifying both elasticity and viscosity from shear waves velocity. To this end, shear wave propagations are simulated in a microscopic cervical tissue model using the finite difference time domain technique, over a wide frequency range from 15 to 200 kHz. Three standard rheological models (Voigt, Maxwell and Zener) are evaluated regarding their ability to reproduce the simulated dispersion curves, and their plausibility to describe cervical tissue is ranked by a stochastic model-class selection formulation. It is shown that the simplest model, i.e. that with less parameters, which best describes the simulated dispersion curves in cervical tissue is the Maxwell model. Furthermore, results show that the excitation frequency determines which rheological model can be representative for the tissue. Typically, viscoelastic parameters tend to converge for excitation frequencies over 100 kHz.

Torsional ultrasonic transducer computational design optimization

A torsional piezoelectric ultrasonic sensor design is proposed in this paper and computationally tested and optimized to measure shear stiffness properties of soft tissue. These are correlated with a number of pathologies like tumors, hepatic lesions and others. The reason is that, whereas compressibility is predominantly governed by the fluid phase of the tissue, the shear stiffness is dependent on the stroma micro-architecture, which is directly affected by those pathologies. However, diagnostic tools to quantify them are currently not well developed. The first contribution is a new typology of design adapted to quasifluids. A second contribution is the procedure for design optimization, for which an analytical estimate of the Robust Probability Of Detection, called RPOD, is presented for use as optimality criteria. The RPOD is formulated probabilistically to maximize the probability of detecting the least possible pathology while minimizing the effect of noise. The resulting optimal transducer has a resonance frequency of 28 kHz.