Defenses

Name: Keshav Rajasekaran

Title: Bio-Inspired flow sensors and their application in Unmanned Air Vehicles

Date: 07/22/2022

Time: 9AM – 11AM

Location: EGR 2164

Committee members:

Professor Miao Yu, Chair

Professor Sarah Bergbreiter, Co-Chair

Professor Don Devoe

Professor Nikhil Chopra 

Professor Ryan Sochol

Professor Pamela Abshire (Dean’s Representative)

Abstract:

Small-scale unmanned air vehicles require lightweight, compact, and low-power sensors that encompass various sensing modalities to enable flight control and navigation in challenging environments. Flow sensing is one such modality that has attracted much interest in recent years. Previously reported flow sensors are mostly fabricated by using the traditional MEMS process and have been primarily used to measure underwater flows.
The overall goal of this dissertation is to develop novel bio-inspired directional flow sensors based on additive manufacturing techniques and explore the application of directional flow sensors for use in micro air vehicles. Three major research thrusts are pursued. First, a micro-scale artificial hair sensor is developed for two-dimensional directional flow sensing. The sensor structure is fabricated by using nano-scale 3D printing, which allows high-precision fabrication with a good device to device uniformity. The performance of the sensor is thoroughly studied in deflection experiments with a probe station and in airflow tests.  
The sensor is integrated with a micro air vehicle (MAV), and detection of flow separation is demonstrated. Second,  flow detection on MAVs with a pair of all elastomer strain sensors is investigated. The soft flow sensors are integrated with an MVA, and the abilities of the sensors for obstacle and gust detection are demonstrated. Finally, the use of bio-inspired flow sensors on a micro air vehicle for performing simple control tasks is explored. The experimental results demonstrate that the sensors are capable of early disturbance warnings, and the sensor output can be used to perform simple navigation tasks, for example, following a wall.

Name: Lena Johnson

Title: Evaluating Flight Time in a Flapping-Wing UAV Through Intermittent Gliding and Flapping

Time: July, 22 4:00-6:00pm EGR-2162 DeWalt Meeting Room

Committee Members: 

– Dr. Hugh Bruck: Committee Chair

– Dr. Derek Paley: Dean’s Representative

– Dr. Miao Yu

– Dr. Yancy Diaz-Mercado

– Dr. Nikhil Chopra

Abstract:

Unmanned Aerial Vehicles (UAVs) are increasingly being used for applications that require longer, reliable flight duration and distances. The greatest limitation to achieving these desired flights is the current on board battery technology which, restricted by internal chemistry and external size, can only provide a finite amount of power over time. Efforts to increase the battery’s efficiency and energy storage tend to rely on cumbersome methods that add weight and/or complexity to the system. However natural flyers, though also limited by a finite amount of internal energy gained through food consumption, are able to extend their flights through techniques that either utilize their inherent aerodynamic advantages or advantageously employ atmospheric phenomena. Flapping-Wing UAVs (FWUAVs) are as limited by their onboard battery as any other type of UAV, but because of their bio-inspired functionality are uniquely suited to utilize natural flight extension methods. Therefore, this PhD presents an analysis of the exploration of bio-inspired, hybrid flapping/gliding, also known as intermittent gliding, techniques to improve the flight performance of a FWUAV. Robo Raven is the FWUAV that was chosen as the research platform for this work. It was developed by researchers at the University of Maryland to perform prolonged, untethered flights and exhibit a flight proficiency that combined the maneuverability of rotary-wing flight with the efficiency of fixed-wing flight.
The technique to improve FWUAV flight time, presented in this work incorporates (1) the modeling of Robo Raven’s flapping/gliding potential through the development of a state-space representation directly linking Robo Raven’s onboard battery dynamics with its aerodynamic performance, (2) the use of the state-space model to characterize the effect of intermittent gliding techniques on flight performance through simulation, (3) the real-world characterization of the simulation and of intermittent gliding techniques through flight demonstrations, and (4) the development of a design space by which the effect of wing design on gliding performance might be explored and lead to the potential tailoring of wing design to desired flight performance. The expected outcome of this technique is scientific analysis of the extension of Robo Raven’s flight time without added complexity of weight of the battery system.

Name: Gyeong Sung Kim

Title of dissertation: Heat and mass transfer analysis and performance improvement study for air gap membrane distillation

Defense Date: 7/21/2022 at 10am

Location: Rm. 4164B, Glenn L. Martin Hall

Zoom link : https://umd.zoom.us/j/3218334573?pwd=TkJFbkozbllBTi9qQlNpOFJGVG12QT09

Committee members: 

Professor Reinhard K. Radermacher, Chair
Professor Kenneth Yu, Dean’s Representative
Professor Jungho Kim
Professor Peter Sandborn
Professor Bao Yang
Research Professor Yunho Hwang

Abstract:

Seawater desalination method can be largely divided into evaporation- and membrane-based techniques. From decades ago, the global installation capacity of reverse-osmosis membrane-based seawater desalination (SWRO) started outgrowing that of the evaporative desalination plant due to its higher energy efficiency and it became the mainstream technology in the 20th century. However, small-scale SWRO facilities installed on South Korean islands are not competitive compared to the thermally driven evaporation method as their specific energy consumption (SEC) values are highly ranging in 9 – 19 kWh∙m^(-3) and there have been frequent maintenance events.

By taking the advantages of direct utilization of renewable and thermal energy, air gap membrane distillation (AGMD) is investigated in this study as an improved approach.  From the preliminary experimental study, it was found that the lower air-gap pressure of AGMD helps to increase its water productivity. However, most of the heat and mass transfer models in AGMD used the constant atmospheric pressure for the air gap.  Therefore, new models considering the pressure effect of the air gap are needed. Since maintaining a vacuum pressure in the gap requires additional energy, a vacuum technique consuming less energy is also needed.  In addition to controlling the total pressure of the gap, condensation augmentation on the cooling surface on one side of the gap is critical since the vapor flux is dependent on the vapor pressure in the gap. As the preliminary experimental study showed that the dropwise condensation mode dominates the condensation of AGMD, the effect of gap size between the condensation surface and hydrophobic membrane is needed to be investigated.

Therefore, this research was performed with the following objectives: (i) experimental investigation and mass transfer model development for vacuum applied AGMD (V-AGMD), (ii) development of a wave-powered desalination system using V-AGMD, (iii) experimental investigation of condensation in AGMD, and (iv) development of condensation enhancement technology for AGMD. From the modeling and experimental research, this study made the following major research outcomes and observations. First, a straightforward mass transfer model was developed by using the concept of Kinetic Theory of Evaporation and temperature fraction value between the fluid temperatures of feed and coolant, based on the AGMD experimental results. This model was evaluated experimentally and showed an excellent prediction of water flux in various air-gap pressures without measuring each temperature of the interface of the feed-membrane-air-cooling surface-coolant. Second, considering that the air gap of AGMD can be operated in a vacuum state using wave power, a novel wave powered AGMD desalination device was proposed and evaluated for island’s dwellers. Third, during the whole AGMD tests, only dropwise condensation (DWC) modes were observed on the stainless-steel condensing wall. Therefore, experiments were conducted to understand the physical pattern of DWC from nucleation to departure. After testing various temperature and humidity conditions, it was confirmed that the average size of the water droplets followed the power law for each case. Fourth, as the periodic cleaning of the condensate wall of AGMD could improve the production of condensate, an experimental study was subsequently performed for the condensation augmentation using an electrohydrodynamic (EHD) method. By both cleaning periodically and applying 2.5 kV and 5.0 kV EHD field on the condensing surface in a thermos-hygrostat chamber, water production rate was increased by 32% and 88%, respectively.

This study concluded that the performance of an AGMD desalination system can be improved by applying a vacuum or an EHD device in its air gap. Therefore, pilot-scale experiments will be conducted as future studies to evaluate the commercial viability of the improved system.

Name: Bahram Parvinian

Defense Date: July 7^th, 2022 at 10am

Location: Glenn L. Martin Hall, EGR-2162

Committee Members:

Professor Jin-Oh Hahn, Advisor/Chair

Professor Yang Tao, Dean’s Representative

Professor Balakumar Balachandran

Professor Yancy Diaz-Mercado

Professor Monifa Vaughn-Cooke

Professor Pras Pathmanathan

Title: A FRAMEWORK FOR CREDIBILITY ASSESSMENT OF SUBJECT-SPECIFIC PHYSIOLOGICAL MODELS 

Abstract:

Physiological closed-loop controllers and decision support systems are medical devices that enable some degree of automation to meet the needs of patients in resource-limited environments such as critical care and surgical units. Traditional methods of safety and effectiveness evidence generation such as pre-clinical animal and human clinical studies are cost prohibitive and may not fully capture different performance attributes of such complex safety-critical systems primarily due to subject variability. In silico studies using subject-specific physiological models (SSPMs) may provide a versatile platform to generate pre-clinical and clinical safety evidence for medical devices and help reduce the size and scope of animal studies and/or clinical trials. To achieve such a goal, the credibility of the SSPMs must be established for the purpose it is intended to serve. While in the past decades significant research has been dedicated towards development of tools and methods for development and evaluation of SSPMs, adoption of such models remains limited, partly due to lack of trust in SSPMs for safety-critical applications. This may be due to a lack of a cohesive and disciplined credibility assessment framework for SSPMs.

In this dissertation a novel framework is proposed for credibility assessment of SSPMs. The framework combines various credibility activities in a unified manner to avoid or reduce resource intensive steps, effectively identify model or data limitations, provide direction as to how to address potential model weaknesses, and provide much needed transparency in the model evaluation process to the decision-makers. To identify various credibility activities, the framework is informed by an extensive literature review of more mature modeling spaces focusing on non-SSPMs as well as a literature review identifying gaps in the published work related to SSPMs. The utility of the proposed framework is successfully demonstrated by its application towards credibility assessment of a CO₂ ventilatory gas exchange model intended to predict physiological parameters, and a blood volume kinetic model intended to predict changes in blood volume in response to fluid resuscitation and hemorrhage. The proposed framework facilitates development of more reliable SSPMs and will result in increased adoption of such models to be used for evaluation of safety-critical medical devices such as Clinical Decision Support (CDS) and Physiological Closed-Loop Controlled (PCLC) systems.

Name: Azin Sadat Mousavi

Defense Date: July 5^th, 2022 at 2 pm

Location: Glenn L. Martin Hall, EGR-2164

Committee Members:

Associate Professor Jin-Oh Hahn, Advisor/Chair

Professor Alison Flatau, Dean’s Representative

Professor Balakumar Balachandran

Professor Hosam Fathy

Professor Miao Yu

Title: Ballistocardiography: Mathematical Modeling, Analysis, and Application to Cardiovascular Health Monitoring

Abstract:

The main goal of this thesis is to improve the early detection and management of cardiovascular disease by developing novel ultra-convenient CV health and risk predictor monitoring techniques based on a physiological signal called ballistocardiogram (BCG). BCG is the recording of heart-induced body movements. It has great potential to enable ultra-convenient CV health monitoring because of its close association with cardiac functions and its amenity for convenient measurement (i.e., measurement form factors including weighing scales and wearables). Nonetheless, the shortage of physical understanding of the BCG is a serious challenge that has hampered its effective use in CV health and risk assessment. The scope of this thesis can be explained under three themes: (i) physics-based modeling of BCG, (ii) BCG recording, and (iii) challenges in wearable BCG-based cuffless blood pressure monitoring.

In the first part of the thesis, a closed-form physics-based model is developed to estimate BCG from a single blood pressure waveform and investigate the feasibility of this model in the estimation of CV risk predictors. This model is inspired by our team’s prior hypothesis that the main mechanism for the genesis of head-to-foot BCG is the pressure gradients in the ascending and descending aorta (the major artery in the body). In addition, a systematic BCG feature selection approach was introduced leveraging the closed-form BCG model. This model-based approach is superior to previous ad-hoc feature selection techniques in that it incorporates physiological knowledge of the arterial system and unlike ad-hoc approaches which are data specific its findings can be generalized to different independent datasets.

BCG waveforms recorded with different sensors and devices have morphological differences. Therefore, the next part of this work is dedicated to the study of different BCG recording devices and the construction of a BCG measurement apparatus that enables the recording of true BCG (as estimated in the mathematical model). The efficacy of the BCG recording apparatus in measuring BCG is shown in two human and animal experiments.

Finally, BCG can enable cuff-less blood pressure (BP) tracking by virtue of two perks. It can easily be instrumented using wearables and it can be used as a proximal timing reference to calculate pulse transit time (PTT) which is the basis of the most common technique for cuff-less BP tracking.  However, most wearable BCG-based studies for cuff-less BP monitoring, have resorted to one posture (standing with hands placed on the sides). Therefore, in this work, the effect of posture on wrist BCG-PPG PTT was investigated. This work revealed the posture-induced changes in PTT and PAT in-depth for the first time, by invoking and quantifying the effect of possible physical mechanisms responsible for such changes.

Name: Sebastian Antonio Romo Duenas

Defense Date: June 21st, 2022 at 1pm.

Location: Glenn Martin Hall, EGR-2162.

Committee Members:

Jelena Srebric, Ph.D. (Advisor/Chair)

Reinhard Radermacher, Ph.D.

Bao Yang, Ph.D.

Dongxia Liu, Ph.D.

Peter Sunderland, Ph.D. (Dean’s Representative)

Title: A Validated Modeling Framework for Performance Analyses of Experimental and Proven Desalination Technologies

Abstract:

There is a wide array of desalination methods available for treating water at different salinities and production rates, but there are no systemic approaches on how to directly compare performance of different desalination systems. Existing comparison efforts focus solely on isolated performance metrics for a single desalination system, resulting in segregated case studies and/or incomparable systems. Numerical models for desalination systems can bridge this gap as they can take account of specific deployment needs. However, models in the literature are not mutually compatible, and they seldom disclose all the parameters or equations necessary for development and validation. This dissertation conceives a cross-comparison enabling simulation framework for the most relevant desalination processes. To achieve this, modeling approaches and thermophysical property correlations are curated from volumes of literature and used to create metamodels for six relevant desalination methods. The models are integrated into a simulation framework based on parameter hierarchies imposed in the model structures. The simulation suite is validated with data from the literature and actual operational data from desalination facilities in the field. 

The results show that the cross-comparison across equal parameter hierarchies is possible for all desalination technologies. A comparative analysis between the dominant technologies in the thermal and molecular transport families, Multi-Effect Distillation (MED) and Reverse Osmosis (RO), respectively, shows that energy intensity in MED is an order of magnitude greater for equivalent operational conditions, but actual operational costs are comparable. The models are further refined to reflect conditions from actual systems in the field and an iterative sampling algorithm is developed to find plausible operation scenarios given the scarce data from the field. This method achieves excellent agreement with data from four desalination plants with percent differences ranging between 2.5% and 9.3%. Furthermore, the results identify two plants performing 20% below their theoretically achievable recovery. Apart from evaluating existing deployments, the simulation suite helps identify a niche in the operational map of existing desalination methods characterized by high recovery rates and high feed salinities that is generally unfulfilled by conventional desalination.

Name: Ali Tivay

Committee Members: 

Professor Jin-Oh Hahn, Chair/Advisor
Professor Perinkulam Krishnaprasad, Dean’s Representative
Professor Hosam Fathy
Professor Nikhil Chopra
Professor Yancy Diaz-Mercado

Date: Thursday, June 16th, 2022

Time: 1:00 PM

NOTE: This defense will take place via Zoom: https://umd.zoom.us/j/3546982448?pwd=ZGFLRUR0MVFlL3B4Wm5PZFgraGxIdz09

Title: Inference-Based Modeling, Monitoring, and Control Algorithms for Autonomous Medical Care

Abstract: Autonomous medical care systems are relatively recent developments in biomedical research that aim to leverage the vigilance, precision, and processing power of computers to assist (or replace) humans in providing medical care to patients.  Indeed, past research has demonstrated initial promise for autonomous medical care in applications related to anesthesia, hemodynamic management, and diabetes management, to name a few.  However, many of these technologies yet do not exhibit the maturity necessary for widespread real-world adoption and regulatory approval.  This can be attributed, in part, to several outstanding challenges associated with the design and development of algorithms that interact with physiological processes: Ideally, an autonomous medical care system should be equipped to exhibit (i) transparent behavior, where the system’s perceptions, reasoning, and decisions are human-interpretable; (ii) context-aware behavior, where the system is capable of remaining mindful of contextual and peripheral information in addition to its primary goal; (iii) coordinated behavior, where the system can coordinate multiple actions in synergistic ways to best achieve multiple objectives; (iv) adaptable behavior, where the system is equipped to identify and adapt to variabilities that exist within and across different patients; and (v) uncertainty-aware behavior, where the system can handle imperfect measurements, quantify the uncertainties that arise as a result, and incorporate them into its decisions.  As these desires and challenges are specific to autonomous medical care applications and not fully explored in past research in this area, this dissertation presents a sequence of methodologies to model, monitor, and control a physiological process with special emphasis on addressing these challenges.  For this purpose, first, a collective variational inference (C-VI) method is presented that facilitates the creation of personalized and generative physiological models from low-information and heterogeneous datasets.  The generative physiological model is of special importance for the purposes of this work, as it encodes physiological knowledge by reproducing the patterned randomness that is observed in physiological datasets.  Second, a population-informed particle filtering (PIPF) method is presented that fuses the information encoded in the generative model with real-time clinical data to form perceptions of a patient’s states, characteristics, and events.  Third, a population-informed variational control (PIVC) method is presented that leverages the generative model, the perceptions of the PIPF algorithm, and user-defined definitions of actions and rewards in order to search for optimal courses of treatment for a patient.  These methods together form a physiological decision-support and closed-loop control (PCLC) framework that is intended to facilitate the desirable behaviors sought in the motivations of this work.  The performance, merits, and limitations of this framework are analyzed and discussed based on a clinically-important case study on fluid resuscitation for hemodynamic management.

Name: Ali Berk Kahraman

Date: June 10th at 10:00 AM; NOTE: This defense will be taking place virtually.

Zoom Link: https://umd.zoom.us/j/4487374796  Meeting ID: 448 737 4796

Committee Members:

Associate Professor Johan Larsson (Chair/Advisor)

Professor James Baeder

Asst. Professor Cristoph Brehm

Professor James Duncan

Professor Arnaud Trouve (Dean’s Representative)

Title of dissertation: Adaptivity in Wall-Modeled Large Eddy Simulation

Abstract: In turbulence-resolving simulations, smaller eddies account for most of the computational cost. This is especially true for a wall-bounded turbulent flow, where a wall-resolved large eddy simulation might use more than 99% of the computing power to resolve the inner 10% of the boundary layer in realistic flows. The solution is to use an approximate model in the inner 10% of the boundary layer where the turbulence is expected to exhibit universal behavior, a technique generally called wall-modeled large eddy simulation. Wall-modeled large-eddy simulation introduces a modeling interface (or exchange location) separating the wall-modeled layer from the rest of the domain. The current state-of-the-art is to rely on user expertise when choosing where to place this modeling interface, whether this choice is tied to the grid or not. This dissertation presents three post-processing algorithms that determine the exchange location systematically.

Two algorithms are physics-based, derived based on known attributes of the turbulence in attached boundary layers. These algorithms are assessed on a range of flows, including flat plate boundary layers, the NASA wall-mounted hump, and different shock/boundary-layer interactions. These algorithms in general agree with what an experienced user would suggest, with thinner wall-modeled layers in nonequilibrium flow regions and thicker wall-modeled layers where the boundary layer is closer to equilibrium, but are completely ignorant to the cost of the simulation they are suggesting.

The third algorithm is based on the sensitivity of the wall-model with the predicted wall shear stress and a model of the subsequent computational cost, finding the exchange location that minimizes a combination of the two. This algorithm is tested both a priori and a posteriori using an equilibrium wall model for the flow over a wall-mounted hump, a boundary layer in an adverse pressure gradient, and a shock/boundary-layer interaction. This third algorithm also produces exchange locations that mostly agree with what an experienced user would suggest, with thinner layers where the wall-model sensitivity is high and thicker layers where this sensitivity is low. This suggests that the algorithm should be useful in simulations of realistic and highly complex geometries.

Name: Parham Dehghani

Title: BURNING EMULATIONS OF CONDENSED PHASE FUELS ABOARD THE INTERNATIONAL SPACE STATION

Committee Members:

Dr. Peter Sunderland, Chair
Dr. James G. Quintiere
Dr. John L. deRis
Dr. Arnaud Trouve
Dr. Reinhard Radermacher
Dr. Christopher Cadou, Dean’s Representative

Date: 6/7/2022
Time: 11:00AM
Location: 3106B, J.M. Patterson Building

Zoom Link: https://umd.zoom.us/j/9827241119

Abstract:

Little is known about the fire hazards of solids and liquids in microgravity. Ground-based tests are too short to overcome ignition transients and testing dozens of condensed fuels in orbit is prohibitively expensive. Burning rate emulation is one way to address this gap. It involves emulating condensed fuels with gases using a porous burner with embedded heat flux gages. This is a study of microgravity burning rate emulation aboard the International Space Station. The burner had porous round surfaces with diameter of 25 mm. The fuel mixture was gaseous ethylene, and it was diluted with various amounts of nitrogen. The resulting heats of combustion were 15 – 47.2 kJ/g. The flow rate, oxygen concentration in the ambient, and pressure were varied. Heat flux to the burner was measured with two embedded heat flux gages and a slug calorimeter. The effective heat of gasification was determined from the heat flux divided by the fuel flow rate. Radiometers provided the radiative loss fractions. A dimensional analysis based on radiation theory yielded a relationship for radiative loss fraction. RADCAL, a narrow-band radiation model, yielded flame emissivities from the product concentrations, temperature, flame length, and pressure. Previously published analytical solutions to these flames allowed prediction of flame heights and radius, and when combined with the radiation empirical relationship led to corrections of total heat release rate from the flames due to radiative loss. Average convective and radiative heat flux were obtained from the analytical solution and a model based on the geometrical view factor of the burner surface with respect to the flame sheet, that were used to calculate heats of gasification. All flames burning in 21% by volume oxygen self-extinguished within 40 s. However, steady flames were observed at 26.5, 34, and 40% oxygen. The analytical solution was used to quantify flame steadiness just before extinction. The steadiest flames reached more than 94% of their steady-state heat fluxes and heights. A flammability map as a plot of heat of gasification versus heat of combustion was developed based on the measurement and theory for nominal ambient oxygen mole fractions of 0.265, 0.34, and 0.4.

Name: Dongyu Chen

Committee Members:

Professor Reinhard Radermacher, Chair/Advisor
Professor Amir Riaz, Co-Chair
Professor Vikrant C. Aute
Professor Jelena Srebric

Professor Siddhartha Das
Professor Bao Yang
Professor Peter B. Sunderland, Dean’s Representative

Date: Tuesday, June 7th, 2022

Time: 1:00 PM

Location: EGR 088-2162 (DeWALT Meeting)

Abstract:

Phase change materials (PCMs) are widely used in thermal energy storage systems, as they can absorb and release a large amount of heat during the phase change process. Numerical simulations can be used for parametric studies and analysis of the thermal performance of the PCM heat exchanger (HX) to produce an optimal design. Among various numerical methods, the lattice Boltzmann method (LBM), a mesoscopic approach that considers the molecular interactions at relatively low computation costs, offers certain key advantages in simulating the phase change process compared with the conventional Navier-Stokes-based (NS-based) methods. Therefore, a comprehensive solid-liquid phase change model is developed based on LBM which is capable of accurately and efficiently simulating the process of convective PCM phase change with and without porous media in both Cartesian and axisymmetric domains. 

Double distribution functions (DDF) coupled with a multi-relaxation-time (MRT) scheme are utilized in the LBM formulation for the simulation of the fluid flow and the temperature field. A differential scanning calorimetry (DSC) correlated equation is applied in LBM to model enthalpy, by which the solid-liquid interface can be automatically tracked. The source term in the MRT scheme is modified to eliminate numerical errors at high Rayleigh numbers. The conjugate thermal model is adopted for the consideration of heat transfer fluid (HTF) flow and conducting fins. Moreover, a parallel computational scheme is used, which allows the model to perform parametric studies efficiently. The new model is verified and validated by various case studies. The results indicate that the new model can successfully predict the process of PCM phase change with errors confined to less than 10%.

Parametric studies are then performed using the validated model to quantitatively evaluate the effect of convection on PCM melting, from which the acceleration rates of PCM melting and the threshold Rayleigh numbers at various aspect ratios are defined and quantified. Furthermore, PCM melting in porous cylindrical HX is also investigated. The results indicate that the acceleration of melting could reach 95% compared to that in pure PCM at 60% energy storage. Moreover, the negative effect of uneven temperature distributions on thermal performance of the HX caused by convection is quantified and analyzed. A modified cylindrical HX that offsets this negative effect by varying the geometry is also evaluated. The results indicate that the modified geometry can successfully enhance heat transfer and balance the uneven temperature distributions.