Convex analysis and unilateral static problems

Dual minimum principles for displacements and stresses are well established for linear variational problems and also for nonlinear (and monotone) constitutive laws. This paper studies the problem of geometric nonlinearity. By introducing a gap function , we recover complementary variational principles in the equilibrium problems of mathematical physics. When the gap function is nonnegative those become minimum principles. The theory is based on convex analysis, and the applications made here are to nonlinear mechanics.

We consider FE-grid optimization in elliptic unilateral boundary value problems. The criterion used in grid optimization is the total potential energy of the system. It is shown that minimization of this cost functional means a decrease of the discretization error or a better approximation of the unilateral boundary conditions. Design sensitivity analysis is given with respect to the movement of nodal points. Numerical results for the Dirichlet-Signorini problem for the Laplace equation and the plane elasticity problem with unilateral boundary conditions are given. In plane elasticity we consider problems with and without friction.

In the sintering of advanced ceramics, a digital twin consisted of a finite element model can predict the final shape and microstructure of a sintered ceramic. However, to minimise uncertainties, it is essential to continually update the mechanical properties in the digital twin using data collected from the manufacturing process. One promising approach for achieving this is through artificial neural networks (ANN). This study introduces a machine learning strategy to update the constitutive behaviour of advanced ceramics in finite element analysis of sintering deformation. A major challenge in implementing machine learning in material processing is the huge amount of data required by the training and validation of an ANN, which are often unavailable or incomplete in a real manufacturing process. This study demonstrates that the data requirement can be reduced by employing a two-step training technique. Firstly, the ANN is trained using a nonlinear constitutive law, which describes a general relationship between the strain rates and stresses. Subsequently, the weights and biases of the ANN are transferred for the retraining using limited experimental data for an actual ceramic. It is shown that such approach can successfully capture the nonlinear constitutive behaviour of fine-grained alumina without demanding for large amount of experimental data. A case study is provided, highlighting the feasibility of implementing the ANN in a commercial finite element package, to replace the constitutive law and predict the shrinkage and distortion of a sintering part. In particular, the sintered dumb-bell shape part, simulated using the retrained ANN, showed grain size and relative density that markedly different from those using the nonlinear constitutive law. It is important to note that the proposed methodology is generic and can be used to create ANNs to replace constitutive laws in finite element analysis in digital twins for a wide range of other engineering processes.

William Thomson, Baron Kelvin of Largs, was the best-known British scientist of his day, who made fundamental contributions to many areas of physics and engineering. Though his life and work have been much studied, his contributions to fluid mechanics have received far less attention that those in heat, electricity, magnetism, geophysics, “ether theory” and telegraphy. After a general introduction, Kelvin’s writings on fluid mechanics are comprehensively surveyed. These reveal the interplay of his mathematical expertise and physical intuition, his deployment of physical analogies, and the origins of some of his work in later-abandoned speculations. Among lasting contributions are his circulation theorem and minimum energy theorem, the misnamed “Stokes’ theorem”, a generalization of Green’s theorem, the method of stationary phase, and much on vortices, instabilities, tides and water waves.

In this contribution three topics are considered. Common to them is a question how to efficiently solve a mechanical problem for a composite material with imperfect bonding between individual material phases. A special attention is paid to the kinematics of the debonding. Namely, to prevent the overlapping of the material phases across the interface where the displacement field is discontinuous. This requirement is expressed as the nonpenetration condition which is a unilateral constraint on the displacements. As a result we have a boundary value problem with unilateral boundary conditions. It is recognized that the simplest way to solve it is to formulate the problem as a variational problem. How can be this done for a time dependent problem is a subject of the first section. To simplify discussion consideration is restricted to a model one dimensional problem. After establishing a stable and consistent numerical method importance of the nonpenetration condition is shown by comparison between solutions with and without imposition of the nonpenetration condition. It is shown that an important step to solve the constrained problem is elimination of the unconstrained degrees of freedom and thus reduction of the problem to the problem with a considerable smaller set of the degrees of freedom. The idea of the reduction of the problem to a smaller set is worth to be pursued on its own.

The application of boundary integral equation methods to potential problems with unilateral boundary condition was developped in [5], [6] and [14].

Time series forecasting is an essential tool across numerous domains, yet traditional models often falter when faced with unilateral boundary conditions, where data is systematically overestimated or underestimated. This paper introduces a novel approach to the task of unilateral boundary time series forecasting. Our research bridges the gap in existing methods by proposing a specialized framework to accurately forecast within these skewed datasets. The cornerstone of our approach is the unilateral mean square error (UMSE), an asymmetric loss function that strategically addresses underestimation biases in training data, improving the precision of forecasts. We further enhance model performance through the implementation of a dual model structure that processes underestimated and accurately estimated data points separately, allowing for a nuanced analysis of the data trends. Additionally, feature reconstruction is employed to recapture obscured dynamics, ensuring a comprehensive understanding of the data. We demonstrate the effectiveness of our methods through extensive experimentation with LightGBM and GRU models across diverse datasets, showcasing superior accuracy and robustness in comparison to traditional models and existing methods. Our findings not only validate the efficacy of our approach but also reveal its model-independence and broad applicability. This work lays the groundwork for future research in this domain, opening new avenues for sophisticated analytical models in various industries where precise time series forecasting is crucial.

In this review article the theoretical foundations for shape-topological sensitivity analysis of elastic energy functional in bodies with nonlinear cracks and inclusions are presented. The results obtained can be used to determine the location and the shape of inclusions which influence in a desirable way the energy release at the crack tip. In contrast to the linear theory, where in principle, crack lips may mutually penetrate, here we employ nonlinear elliptic boundary value problems in non-smooth domains with cracks with non-penetration contact conditions across the crack lips or faces. A shape-topological sensitivity analysis of the associated variational inequalities is performed for the elastic energy functional. Topological derivatives of integral shape functionals for variational inequalities with unilateral boundary conditions are derived. The closed form results are obtained for the Laplacian and linear elasticity in two and three spatial dimensions. Singular geometrical perturbations in the form of cavities or inclusions are considered. In the variational context the singular perturbations are replaced by regular perturbations of bilinear forms. The obtained expressions for topological derivatives are useful in numerical method of shape optimization for contact problems as well as in passive control of crack propagation.

On considere des problemes de Stefan a 1 phase a 1 dimension avec la condition limite unilaterale sur la frontiere fixee

Helical buckling and postbuckling of pre-stressed cylindrical tubes under finite torsion

Improved nonlinear rotational constitutive law for predicting critical loads in elastomeric isolation bearings

The general criterion of fracture considered represents a modification of the Griffith theory. It is shown that Rice's criterion is a particular case of the general criterion. The theorem of minimum complementary energy is examined along with questions regarding the formation of a crack, taking into account irreversible effects such as plastic energy dissipation

BIEM-based variational principles for elastoplasticity with unilateral contact boundary conditions

Artificial neural network (ANN) models are used to learn the nonlinear constitutive laws based on indirectly measurable data. The real input and output of the ANN model are derived from indirect data using a mechanical system, which is composed of several subsystems including the ANN model. As the ANN model is coupled with other subsystems, the input of the ANN model needs to be determined during the training. This approach integrates measurable data, mechanics, and ANN models so that the ANN models can be trained without direct data which is usually not available from experiments. Two examples are provided as an illustration of the proposed approach. The first example uses two-dimensional (2D) finite element (FE) analysis to train an ANN model to learn the nonlinear in-plane shear constitutive law. The second example applies a continuum damage model to train an ANN model to learn the damage accumulation law. The results show that the trained ANN models achieve great accuracy based on the proposed approach.

Variability response functions for beams with nonlinear constitutive laws