arXiv subjects

Alessandro Reali

Publications and source records attributed to Alessandro Reali.

At least 37 records · Page 2Linked to original sources

arXiv2021

Coupled and Uncoupled Dynamic Mode Decomposition in Multi-Compartmental Systems with Applications to Epidemiological and Additive Manufacturing Problems

Dynamic Mode Decomposition (DMD) is an unsupervised machine learning method that has attracted considerable attention in recent years owing to its equation-free structure, ability to easily identify coherent spatio-temporal structures in data, and effectiveness in providing reasonably accurate predictions for certain problems. Despite these successes, the application of DMD to certain problems featuring highly nonlinear transient dynamics remains challenging. In such cases, DMD may not only fail to provide acceptable predictions but may indeed fail to recreate the data in which it was trained, restricting its application to diagnostic purposes. For many problems in the biological and physical sciences, the structure of the system obeys a compartmental framework, in which the transfer of mass within the system moves within states. In these cases, the behavior of the system may not be accurately recreated by applying DMD to a single quantity within the system, as proper knowledge of the system dynamics, even for a single compartment, requires that the behavior of other compartments is taken into account in the DMD process. In this work, we demonstrate, theoretically and numerically, that, when performing DMD on a fully coupled PDE system with compartmental structure, one may recover useful predictive behavior, even when DMD performs poorly when acting compartment-wise. We also establish that important physical quantities, as mass conservation, are maintained in the coupled-DMD extrapolation. The mathematical and numerical analysis suggests that DMD may be a powerful tool when applied to this common class of problems. In particular, we show interesting numerical applications to a continuous delayed-SIRD model for Covid-19, and to a problem from additive manufacturing considering a nonlinear temperature field and the resulting change of material phase from powder, liquid, and solid states.

cs.LG↗

arXiv2021

Dynamic Mode Decomposition in Adaptive Mesh Refinement and Coarsening Simulations

Dynamic Mode Decomposition (DMD) is a powerful data-driven method used to extract spatio-temporal coherent structures that dictate a given dynamical system. The method consists of stacking collected temporal snapshots into a matrix and mapping the nonlinear dynamics using a linear operator. The standard procedure considers that snapshots possess the same dimensionality for all the observable data. However, this often does not occur in numerical simulations with adaptive mesh refinement/coarsening schemes (AMR/C). This paper proposes a strategy to enable DMD to extract features from observations with different mesh topologies and dimensions, such as those found in AMR/C simulations. For this purpose, the adaptive snapshots are projected onto the same reference function space, enabling the use of snapshot-based methods such as DMD. The present strategy is applied to challenging AMR/C simulations: a continuous diffusion-reaction epidemiological model for COVID-19, a density-driven gravity current simulation, and a bubble rising problem. We also evaluate the DMD efficiency to reconstruct the dynamics and some relevant quantities of interest. In particular, for the SEIRD model and the bubble rising problem, we evaluate DMD's ability to extrapolate in time (short-time future estimates).

cs.LG↗

arXiv2021

Image-based numerical characterization and experimental validation of tensile behavior of octet-truss lattice structures

The production of lightweight metal lattice structures has received much attention due to the recent developments in additive manufacturing (AM). The design flexibility comes, however, with the complexity of the underlying physics. In fact, metal additive manufacturing introduces process-induced geometrical defects that mainly result in deviations of the effective geometry from the nominal one. This change in the final printed shape is the primary cause of the gap between the as-designed and as-manufactured mechanical behavior of AM products. Thus, the possibility to incorporate the precise manufactured geometries into the computational analysis is crucial for the quality and performance assessment of the final parts. Computed tomography (CT) is an accurate method for the acquisition of the manufactured shape. However, it is often not feasible to integrate the CT-based geometrical information into the traditional computational analysis due to the complexity of the meshing procedure for such high-resolution geometrical models and the prohibitive numerical costs. In this work, an embedded numerical framework is applied to efficiently simulate and compare the mechanical behavior of as-designed to as-manufactured octet-truss lattice structures. The parts are produced using laser powder bed fusion (LPBF). Employing an immersed boundary method, namely the Finite Cell Method (FCM), we perform direct numerical simulations (DNS) and first-order numerical homogenization analysis of a tensile test for a 3D printed octet-truss structure. Numerical results based on CT scan (as-manufactured geometry) show an excellent agreement with experimental measurements, whereas both DNS and first-order numerical homogenization performed directly on the 3D virtual model (as-designed geometry) of the structure show a significant deviation from experimental data.

cs.CE↗

arXiv2021

Mixed variational formulations for structural topology optimization based on the phase-field approach

We propose a variational principle combining a phase-field functional for structural topology optimization with a mixed (three-field) Hu-Washizu functional, then including directly in the formulation equilibrium, constitutive, and compatibility equations. The resulting mixed variational functional is then specialized to derive a classical topology optimization formulation (where the amount of material to be distributed is an \emph{a priori} assigned quantity acting as a global constraint for the problem) as well as a novel topology optimization formulation (where the amount of material to be distributed is minimized, hence with no pre imposed constraint for the problem). Both formulations are numerically solved by implementing a mixed finite element scheme, with the second approach avoiding the introduction of a global constraint, hence respecting the convenient local nature of the finite element discretization. Furthermore, within the proposed approach it is possible to obtain guidelines for settings proper values of phase-field-related simulation parameters and, thanks to the combined phase-field and Hu-Washizu rationale, a monolithic algorithm solution scheme can be easily adopted. An insightful and extensive numerical investigation results in a detailed convergence study and a discussion on the obtained final designs. The numerical results clearly highlight differences between the two formulations as well as advantages related to the monolithic solution strategy; numerical investigations address both two-dimensional and three-dimensional applications.

math.NA↗

arXiv2021

Explicit high-order generalized-$α$ methods for isogeometric analysis of structural dynamics

We propose a new family of high-order explicit generalized-$α$ methods for hyperbolic problems with the feature of dissipation control. Our approach delivers $2k,\, \left(k \in \mathbb{N}\right)$ accuracy order in time by solving $k$ matrix systems explicitly and updating the other $2k$ variables at each time-step. The user can control the numerical dissipation in the discrete spectrum's high-frequency regions by adjusting the method's coefficients. We study the method's spectrum behaviour and show that the CFL condition is independent of the accuracy order. The stability region remains invariant while we increase the accuracy order. Next, we exploit efficient preconditioners for the isogeometric matrix to minimize the computational cost. These preconditioners use a diagonal-scaled Kronecker product of univariate parametric mass matrices; they have a robust performance with respect to the spline degree and the mesh size, and their decomposition structure implies that their application is faster than a matrix-vector product involving the fully-assembled mass matrix. Our high-order schemes require simple modifications of the available implementations of the generalized-$α$ method. Finally, we present numerical examples demonstrating the methodology's performance regarding single- and multi-patch IGA discretizations.

math.NA↗

arXiv2021

Assessing the spatio-temporal spread of COVID-19 via compartmental models with diffusion in Italy, USA, and Brazil

The outbreak of COVID-19 in 2020 has led to a surge in interest in the mathematical modeling of infectious diseases. Such models are usually defined as compartmental models, in which the population under study is divided into compartments based on qualitative characteristics, with different assumptions about the nature and rate of transfer across compartments. Though most commonly formulated as ordinary differential equation (ODE) models, in which the compartments depend only on time, recent works have also focused on partial differential equation (PDE) models, incorporating the variation of an epidemic in space. Such research on PDE models within a Susceptible, Infected, Exposed, Recovered, and Deceased (SEIRD) framework has led to promising results in reproducing COVID-19 contagion dynamics. In this paper, we assess the robustness of this modeling framework by considering different geometries over more extended periods than in other similar studies. We first validate our code by reproducing previously shown results for Lombardy, Italy. We then focus on the U.S. state of Georgia and on the Brazilian state of Rio de Janeiro, one of the most impacted areas in the world. Our results show good agreement with real-world epidemiological data in both time and space for all regions across major areas and across three different continents, suggesting that the modeling approach is both valid and robust.

q-bio.PE↗

arXiv2021

Bending behavior of additively manufactured lattice structures: numerical characterization and experimental validation

Selective Laser Melting (SLM) technology has undergone significant development in the past years providing unique flexibility for the fabrication of complex metamaterials such as octet-truss lattices. However, the microstructure of the final parts can exhibit significant variations due to the high complexity of the manufacturing process. Consequently, the mechanical behavior of these lattices is strongly dependent on the process-induced defects, raising the importance on the incorporation of as-manufactured geometries into the computational structural analysis. This, in turn, challenges the traditional mesh-conforming methods making the computational costs prohibitively large. In the present work, an immersed image-to-analysis framework is applied to efficiently evaluate the bending behavior of AM lattices. To this end, we employ the Finite Cell Method (FCM) to perform a three-dimensional numerical analysis of the three-point bending test of a lattice structure and compare the as-designed to as-manufactured effective properties. Furthermore, we undertake a comprehensive study on the applicability of dimensionally reduced beam models to the prediction of the bending behavior of lattice beams and validate classical and strain gradient beam theories applied in combination with the FCM. The numerical findings suggest that the SLM octet-truss lattices exhibit size effects, thus, requiring a flexible framework to incorporate high-order continuum theories.

cs.CE↗

arXiv2020

Efficient equilibrium-based stress recovery for isogeometric laminated curved structures

This work focuses on an efficient stress recovery procedure for laminated composite curved structures, which relies on Isogeometric Analysis (IGA) and equilibrium. Using a single element through the thickness in combination with a calibrated layerwise integration rule or a homogenized approach, the 3D solid isogeometric modeling grants an inexpensive and accurate approximation in terms of displacements (and their derivatives) and in-plane stresses, while through-the-thickness stress components are poorly approximated. Applying a further post-processing step, an accurate out-of-plane stress state is also recovered, even from a coarse displacement solution. This is based on a direct integration of the equilibrium equations in strong form, involving high order derivatives of the displacement field. Such a continuity requirement is fully granted by IGA shape function properties. The post-processing step is locally applied, which grants that no additional coupled terms appear in the equilibrium, allowing for a direct reconstruction without the need to further iterate to resolve the out-of-balance momentum equation. Several numerical results show the good performance of this approach particularly for composite stacks with significant radius-to-thickness ratio and number of plies. In particular, in the latter case, where a layerwise technique employing a number of degrees of freedom directly proportional to the number of plies would be much more computationally demanding, the proposed method can be regarded as a very appealing alternative choice.

math.NA↗

arXiv2020

Accurate equilibrium-based interlaminar stress recovery for isogeometric laminated composite Kirchhoff plates

In this paper, we use isogeometric Kirchhoff plates to approximate composite laminates adopting the classical laminate plate theory. Both isogeometric Galerkin and collocation formulations are considered. Within this framework, interlaminar stresses are recovered through an effective post-processing technique based on the direct imposition of equilibrium in strong form, relying on the accuracy and the higher continuity typically granted by isogeometric discretizations. The effectiveness of the proposed approach is proven by extensive numerical tests.

math.NA↗

arXiv2020

Optimal control of cytotoxic and antiangiogenic therapies on prostate cancer growth

Prostate cancer can be lethal in advanced stages, for which chemotherapy may become the only viable therapeutic option. While there is no clear clinical management strategy fitting all patients, cytotoxic chemotherapy with docetaxel is currently regarded as the gold standard. However, tumors may regain activity after treatment conclusion and become resistant to docetaxel. This situation calls for new delivery strategies and drug compounds enabling an improved therapeutic outcome. Combination of docetaxel with antiangiogenic therapy has been considered a promising strategy. Bevacizumab is the most common antiangiogenic drug, but clinical studies have not revealed a clear benefit from its combination with docetaxel. Here, we capitalize on our prior work on mathematical modeling of prostate cancer growth subjected to combined cytotoxic and antiangiogenic therapies, and propose an optimal control framework to robustly compute the drug-independent cytotoxic and antiangiogenic effects enabling an optimal therapeutic control of tumor dynamics. We describe the formulation of the optimal control problem, for which we prove the existence of at least a solution and determine the necessary first order optimality conditions. We then present numerical algorithms based on isogeometric analysis to run a preliminary simulation study over a single cycle of combined therapy. Our results suggest that only cytotoxic chemotherapy is required to optimize therapeutic performance and we show that our framework can produce superior solutions to combined therapy with docetaxel and bevacizumab. We also illustrate how the optimal drug-naïve cytotoxic effects computed in these simulations may be successfully leveraged to guide drug production and delivery strategies by running a nonlinear least-square fit of protocols involving docetaxel and a new design drug.

math.OC↗

arXiv2020

Reduced Order Isogeometric Analysis Approach for PDEs in Parametrized Domains

In this contribution, we coupled the isogeometric analysis to a reduced order modelling technique in order to provide a computationally efficient solution in parametric domains. In details, we adopt the free-form deformation method to obtain the parametric formulation of the domain and proper orthogonal decomposition with interpolation for the computational reduction of the model. This technique provides a real-time solution for any parameter by combining several solutions, in this case computed using isogeometric analysis on different geometrical configurations of the domain, properly mapped into a reference configuration. We underline that this reduced order model requires only the full-order solutions, making this approach non-intrusive. We present in this work the results of the application of this methodology to a heat conduction problem inside a deformable collector pipe.

math.NA↗

arXiv2020

Simulating the spread of COVID-19 via spatially-resolved susceptible-exposed-infected-recovered-deceased (SEIRD) model with heterogeneous diffusion

We present an early version of a Susceptible-Exposed-Infected-Recovered-Deceased (SEIRD) mathematical model based on partial differential equations coupled with a heterogeneous diffusion model. The model describes the spatio-temporal spread of the COVID-19 pandemic, and aims to capture dynamics also based on human habits and geographical features. To test the model, we compare the outputs generated by a finite-element solver with measured data over the Italian region of Lombardy, which has been heavily impacted by this crisis between February and April 2020. Our results show a strong qualitative agreement between the simulated forecast of the spatio-temporal COVID-19 spread in Lombardy and epidemiological data collected at the municipality level. Additional simulations exploring alternative scenarios for the relaxation of lockdown restrictions suggest that reopening strategies should account for local population densities and the specific dynamics of the contagion. Thus, we argue that data-driven simulations of our model could ultimately inform health authorities to design effective pandemic-arresting measures and anticipate the geographical allocation of crucial medical resources.

q-bio.PE↗

arXiv2019

Error-estimate-based Adaptive Integration For Immersed Isogeometric Analysis

The Finite Cell Method (FCM) together with Isogeometric analysis (IGA) has been applied successfully in various problems in solid mechanics, in image-based analysis, fluid-structure interaction and in many other applications. A challenging aspect of the isogeometric finite cell method is the integration of cut cells. In particular in three-dimensional simulations the computational effort associated with integration can be the critical component of a simulation. A myriad of integration strategies has been proposed over the past years to ameliorate the difficulties associated with integration, but a general optimal integration framework that suits a broad class of engineering problems is not yet available. In this contribution we provide a thorough investigation of the accuracy and computational effort of the octree integration scheme. We quantify the contribution of the integration error using the theoretical basis provided by Strang's first lemma. Based on this study we propose an error-estimate-based adaptive integration procedure for immersed isogeometric analysis. Additionally, we present a detailed numerical investigation of the proposed optimal integration algorithm and its application to immersed isogeometric analysis using two- and three-dimensional linear elasticity problems.

math.NA↗

arXiv2019

A simple and effective method based on strain projections to alleviate locking in isogeometric solid shells

In this work, we focus on the family of shell formulations referred to as "solid shells", where the simulation of shell-type structures is performed by means of a mesh of 3D solid elements, with typically only one element through the thickness. We propose a novel approach for alleviating the various locking phenomena, which typically appear in thin structures, based on the projection of strains onto discontinuous coarser polynomial spaces defined at element level. In particular, we present and investigate two different formulations based on this approach. Several numerical experiments prove the very good performance of both formulations. The main advantages of the presented approach compared to existing solid shell formulations are its simplicity and numerical efficiency.

math.NA↗

arXiv2019

Mathematical analysis and simulation study of a phase-field model of prostate cancer growth with chemotherapy and antiangiogenic therapy effects

Cytotoxic chemotherapy is a common treatment for advanced prostate cancer. These tumors are also known to rely on angiogenesis, i.e., the growth of local microvasculature via chemical signaling produced by the tumor. Thus, several clinical studies have been investigating antiangiogenic therapy for advanced prostate cancer, either as monotherapy or combined with standard cytotoxic protocols. However, the complex genetic alterations promoting prostate cancer growth complicate the selection of the best chemotherapeutic approach for each patient's tumor. Here, we present a mathematical model of prostate cancer growth and chemotherapy that may enable physicians to test and design personalized chemotherapeutic protocols in silico. We use the phase-field method to describe tumor growth, which we assume to be driven by a generic nutrient following reaction-diffusion dynamics. Tumor proliferation and apoptosis (i.e., programmed cell death) can be parameterized with experimentally-determined values. Cytotoxic chemotherapy is included as a term downregulating tumor net proliferation, while antiangiogenic therapy is modeled as a reduction in intratumoral nutrient supply. Another equation couples the tumor phase field with the production of prostate-specific antigen, which is an extensively used prostate cancer biomarker. We prove the well-posedness of our model and we run a series of representative simulations using an isogeometric method to explore untreated tumor growth as well as the effects of cytotoxic chemotherapy and antiangiogenic therapy, both alone and combined. Our simulations show that our model captures the growth morphologies of prostate cancer as well as common outcomes of cytotoxic and antiangiogenic mono and combined therapy. Our model also reproduces the usual temporal trends in tumor volume and prostate-specific antigen evolution observed in previous studies.

math.AP↗

arXiv2019

Structural multiscale topology optimization with stress constraint for additive manufacturing

In this paper a phase-field approach for structural topology optimization for a 3D-printing process which includes stress constraint and potentially multiple materials or multiscales is analyzed. First order necessary optimality conditions are rigorously derived and a numerical algorithm which implements the method is presented. A sensitivity study with respect to some parameters is conducted for a two-dimensional cantilever beam problem. Finally, a possible workflow to obtain a 3D-printed object from the numerical solutions is described and the final structure is printed using a fused deposition modeling (FDM) 3D printer.

math.OC↗

arXiv2019

Graded-material Design based on Phase-field and Topology Optimization

In the present work we introduce a novel graded-material design based on phase-field and topology optimization. The main novelty of this work comes from the introduction of an additional phase-field variable in the classical single-material phase-field topology optimization algorithm. This new variable is used to grade the material properties in a continuous fashion. Two different numerical examples are discussed, in both of them we perform sensitivity studies to asses the effects of different model parameters onto the resulting structure. From the presented results we can observe that the proposed algorithm adds additional freedom in the design, exploiting the higher flexibility coming from additive manufacturing technology.

math.OC↗

arXiv2019

Accurate prediction of melt pool shapes in laser powder bed fusion by the non-linear temperature equation including phase changes - isotropic versus anisotropic conductivity

In this contribution, we validate a physical model based on a transient temperature equation (including latent heat) w.r.t. the experimental set AMB2018-02 provided within the additive manufacturing benchmark series, established at the National Institute of Standards and Technology, USA. We aim at predicting the following quantities of interest: width, depth, and length of the melt pool by numerical simulation and report also on the obtainable numerical results of the cooling rate. We first assume the laser to posses a double ellipsoidal shape and demonstrate that a well calibrated, purely thermal model based on isotropic thermal conductivity is able to predict all the quantities of interest, up to a deviation of maximum 7.3\% from the experimentally measured values. However, it is interesting to observe that if we directly introduce, whenever available, the measured laser profile in the model (instead of the double ellipsoidal shape) the investigated model returns a deviation of 19.3\% from the experimental values. This motivates a model update by introducing anisotropic conductivity, which is intended to be a simplistic model for heat material convection inside the melt pool. Such an anisotropic model enables the prediction of all quantities of interest mentioned above with a maximum deviation from the experimental values of 6.5\%. We note that, although more predictive, the anisotropic model induces only a marginal increase in computational complexity.

cs.CE↗