PEMesh: a Graphical Framework for the Analysis of the InterplayBetween Geometry and PEM Solvers
PPEMesh : a Graphical Framework for the Analysis of the InterplayBetween Geometry and PEM Solvers ⋆ Daniela Cabiddu a , ∗ ,1 , Giuseppe Patanè a ,2 and Michela Spagnuolo a ,3 a CNR-IMATI, Genova, Italy
A R T I C L E I N F O
Keywords :Computer-Aided Geometric DesignGraphical User InterfacePolytonal meshesPartial Differential EquationsPolytopal Element Methods (PEM)
A B S T R A C T
Partial differential equations can be solved on general polygonal and polyhedral meshes, through
Poly-topal Element Methods (PEMs). Unfortunately, the relation between geometry and analysis is stillunknown and subject to ongoing research in order to identify weaker shape-regularity criteria underwhich PEMs can reliably work. We propose
PEMesh , a graphical framework to support the analysis ofthe relation between the geometric properties of polygonal meshes and the numerical performances ofPEM solvers.
PEMesh allows the design of polygonal meshes that increasingly stress some geometricproperties, by exploiting any external PEM solver, and supports the study of the correlation betweenthe performances of such a solver and geometric properties of the input mesh. Furthermore, it ishighly modular, customisable, easy to use, and provides the possibility to export analysis results bothas numerical values and graphical plots.
PEMesh has a potential practical impact on ongoing and futureresearch activities related to PEM methods, polygonal mesh generation and processing.
1. Introduction
Over the last fifty years, computer-based simulation hasdramatically increased its impact on research, design, andproduction, and is now an indispensable tool for develop-ment and innovation in science and technology. In particu-lar, Partial Differential Equations (PDEs) offer a broad andflexible framework for modeling and analyzing a number ofphenomena arising in fields as diverse as physics, engineer-ing, biology, and medicine. Computer-based simulation ofPDEs also relies on a suitable description of geometrical en-tities, such as the computational domain and its properties.However, the representation of geometric entities has beenstudied mainly in the field of geometric modeling, and of-ten the requirements of shape design are different from thoseones of numerical simulation.In this context,
Polytopal Element Methods (PEMs) al-low solving differential equations on general polygonal andpolyhedral meshes, thus offering a great freedom in the defi-nition of mesh generation algorithms. Similarly to
Finite El-ements Methods (FEMs), the performance of PEMs (i.e., ac-curacy, stability, effectiveness of preconditioning) dependson the quality of the underlying mesh. Differently from FEMs,where the relation between the geometric properties of themesh and the performances of the solver are well known [22,6, 3], the definition of the quality of polytopal elements isstill subject to ongoing research [11, 14, 2].The proposed graphical framework is intended to sup-port the analysis of the relation between the geometric prop- ⋆ This document is the results of the research project funded by the ERCProject CHANGE, which has received funding from the European ResearchCouncil (ERC) under the European Union, Horizon 2020 research and in-novation programme (grant agreement Nr. 694515). ∗ Corresponding author [email protected] (D. Cabiddu); [email protected] (G. Patanè); [email protected] (M. Spagnuolo)
ORCID (s): (D. Cabiddu) erties of the mesh and the numerical performances of thesolver, in terms of basis degree, conditioning number of thestiffness matrix, etc.. To this end, our work covers severalaspects, such as the design and generation of meshes to in-creasingly stress geometric properties, the study of the per-formances of PEM solvers, and the correlation between suchperformances and main geometric properties of the inputmeshes. Each step is performed by exploiting existing toolsmainly coming from two related but independent researchareas: geometric design and numerical methods for PEMs.Indeed, these tools rely on different representations of thesame domain, and researchers are often required to be skilledprogrammers and expert tool users to allow such tools to bepart of the same experimental pipeline.
Figure 1:
Main window of
PEMesh . We introduce
PEMesh (Fig. 1), as an open-source softwaretool designed to help researchers to perform experiments onthe analysis and design of polytopal meshes for PEM solvers.
PEMesh is an advanced graphical tool that seamless integratesgeometric design pipelines and PEM simulations. It sup-ports the design and generation of complex input polygonal
D. Cabiddu, G. Patanè, M. Spagnuolo:
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Page 1 of 7 a r X i v : . [ c s . C G ] F e b EMesh : a Graphical Framework for the Analysis of the Interplay Between Geometry and PEM Solvers meshes by stressing geometric properties, while providingthe possibility to solve PEMs on the generated meshes. Fur-thermore,
PEMesh allows the user to correlate one or moregeometric properties of the input polytopal mesh with theperformances of PEM solvers, and to visualize the resultsthough customisable and interactive plots.
PEMesh is highly modular and customisable. It allows re-searchers to simulate any PEM solver, by simply calling thePEM solver executable from an internal command line andproviding possibly additional input parameters other than thegeometric data set. To the best of our knowledge,
PEMesh isthe first graphical tool to generate complex discrete polytopalmeshes and to support a study of the correlation betweentheir geometric properties and numerical PEMs solvers. In-deed, it has a potential practical impact on research activitieson this subject.The paper is organised as follows. We briefly reviewprevious work on PEMs and existing tools both to performPEM simulations and to design geometric data sets (Sect. 2).Then, we describe the structure of
PEMesh and its capabili-ties (Sect. 3), with technical details about its implementa-tion (Sect. 4). Finally, we discuss some directions of futureresearch (Sect. 5).
2. Background and related work
We briefly review previous work on numerical FEM solversand libraries (Sect. 2.1) and meshing tools (Sect. 2.2).
Main PEMs include Mimetic Finite Differences [27, 4],Discontinuous Galerkin-Finite Element Method (DG-FEM) [1,5], Hybridisable and Hybrid High-Order Methods [7, 8], WeakGalerkin Method [28], BEM-based FEM [19], Poly-SplineFEM [20], and Polygonal FEM [23]. Main existing tools forthe numerical solution of PDEs include (i)
VEMLab [17],which is an open source MATLAB library for the virtual ele-ment method and (ii)
Veamy , which is a free and open sourceC++ library that implements the virtual element method (c++version of [17]). The current release of this library allowsthe solution of 2D linear elasto-static problems and the 2DPoisson problem [18]. Other libraries are (iii) the 50-linesMATLAB implementation of the lowest order virtual ele-ment method for the two-dimensional Poisson problem ongeneral polygonal meshes [24], and (iv) the MATLAB im-plementation of the lowest order Virtual Element Method(VEM) [13].As a matter of example, we demonstrate how a PEMsolver can be integrated in
PEMesh . Our use case exploits the
Virtual Element Method (VEM) [26], which can be consid-ered as an extension to FEM for handling general polytopalmeshes.
Nowadays, meshes are commonplace in a number of ap-plications ranging from engineering to bio-medicine and ge-ology. Depending on the application field, automatic mesh generation may be a difficult task, due to specific geometricrequirements to be satisfied.With reference to simulation with FEMs, the principlebehind meshing algorithms in commercial FEM solvers aredescribed in [15], and an open-source tool is provided. Free-FEM [10] is a popular 2D and 3D partial differential equa-tions (PDE) solver used by thousands of researchers acrossthe world, including its own mesh generation module. Al-though it provides plenty of functionalities, it is based on itsown language and it has no graphical interface. Similarly,the MATLAB ® suite provides its own FEM mesh genera-tor [16]. Both solutions focus on FEM requirements, andenable the possibility to generate triangle meshes, but theydo not allow the generation of generic polygon meshes.Concerning polygonal meshes, available Voronoi-basedmeshing tools (e.g. [9, 25]) are not suited for our study, be-cause they produce convex elements that are not challengingenough to stress PEM solvers. For the best of our knowledge,the benchmark proposed in [2] is the only one providing apolygonal mesh generation approach specifically designedfor PDE solvers. Unfortunately, the proposed approach isnot easily customisable, and allows the generation of poly-gon meshes having a single non-triangle element. PEMesh provides an advanced mesh generation modulewhich enables the creation of generic polygonal meshes specif-ically designed for PDE solvers.
3. Proposed framework
PEMesh is aimed at evaluating the dependence of the per-formances of a PEM solver on geometrical properties of theinput polygonal mesh, which is either generated by using
PEMesh itself or provided as an external resource.
PEMesh ismainly composed by four modules, each of them is providedas a specialized window.
Polygon mesh generation & loading allows the user toload one or more existing meshes or to generate a new onesfrom scratch by either exploiting a set of provided polytopalelements or providing an external one. The generation ofnew meshes is highly customisable, and the user is allowedto play with a large set of options and parameters (Sect. 3.1). Geometric analysis allows the user to perform a deepanalysis of geometric properties of the input polygonal meshesand to correlate each of them with the others. Results of suchan analysis are shown though advanced plots (Sect. 3.2).
PEM solver allows the user to run a PEM solver and toanalyse its performances on input polygonal meshes. AnyPEM solver may be exploited, as long as it can be run fromcommand line and provides its output according to a specifictextual format. Both the solution and the ground-throughof the PEM is shown directly on the meshes, while the per-formances of the solver are visualized through linear plots(Sect. 3.3).
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EMesh : a Graphical Framework for the Analysis of the Interplay Between Geometry and PEM Solvers
Figure 2:
Meshes generated by selecting the same set of poly-gons, but editing each element differently.
Correlation visualization supports the analysis of the cor-relation between geometric properties of the polygonal meshesand numerical performances of the selected PEM solver. Re-sults are made available in the form of customizable scatterplots (Sect. 3.4).
PEMesh provides the possibility to show the results of eachanalysis step on the display, to customize visualization as-pects of plots (i.e., color, font sizes, etc.) and to interactivelyanalyze them by clicking on visualised points and lines toshow data values in the selected point. Furthermore, suchresults can be saved on disk as images and as textual files, tobe possibly re-used by other applications. & loading This module is the main core of
PEMesh and it is startedas soon as the application is run. It provides the possibilityto load an existing polygonal mesh or to generate a new onefrom scratch. In
PEMesh , the mesh generation method takes acue from the approach described in [2], where the domain issupposed to be a squared canvas, and the area of the domainwhich is not covered by a polygon is filled with trianglesusing [21].Differently from [2],
PEMesh supports the generation ofmeshes with more than one non-triangle polygon, whose po-sition, scale and rotation is chosen by the user before apply-ing the triangulation of the external domain (Fig. 2). Fur-thermore, triangulation parameters are set according to theuser needs (e.g., fixing the area of the triangles or the mini-mum angle) and the mirroring approach proposed in [2] canbe eventually applied after the triangulation.An additional feature is the aggregation of the generatedtriangles to create generic polygons (Fig. 3). This feature al-lows the generation of generic polygonal meshes where thenumber of triangles is reduced almost to zero and some geo-metric properties are stressed all over the discretised domain.The aggregation criterion guarantees that the diameter of thepolygons generated by aggregation is at most equal to the di-ameter of the smallest user-selected polygon.
When one or more polygonal meshes are available, eithergenerated from scratch or loaded from disk,
PEMesh allows adeep geometric analysis and provides a visual summary of
Figure 3:
Mesh with aggregated triangles. geometric properties. Specifically, our approach considers aset of polygonal metrics (Table 1, Fig. 4), also consideringtheir minimum, maximum and average values. (a) (b)
Figure 4: (a) Geometric metrics of a polygon and (b) theircounterpart on a polygonal mesh. Note that 𝑚𝑖𝑛 ( 𝐾𝐸 ) =0 and it corresponds to the central non-triangle polygon,while 𝑚𝑖𝑛 ( 𝑀𝑃 𝐷 ) = 𝑚𝑖𝑛 ( 𝑆𝐸 ) and 𝑚𝑎𝑥 ( 𝑀𝑃 𝐷 ) = 𝑚𝑎𝑥 ( 𝑆𝐸 ) . Our polygonal metrics are classified into 6 main classes:• edges : number of edges (nE) of the input polygon,shortest edge (SE);• angles : ratio MA/mA, with MA, mA minimum, max-imum inner angle of the polygon, respectively;• areas : area (AR) of the polygon, kernel area (KE),kernel-area ratio (i.e., ratio between the area of the ker-nel of the polygon and its whole area), area-perimeterratio (APR);• radii : inscribed circle radius (IC), circumscribed cir-cle radius (CC), circle ratio (CR:=IC/CC);• distances : minimum point to point distance (MPD),normalized point distance (NPD) (i.e., normalized ver-sion of MPD);• shape regularity (SRG), as ratio between the radiusof the circle to the circumscribed to the polygon andthe radius of the circle inscribed in the kernel of theelement.
PEMesh provides three different visualizations of the geomet-ric analysis, thus enabling the possibility to either analyze
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EMesh : a Graphical Framework for the Analysis of the Interplay Between Geometry and PEM Solvers
Table 1
Proposed polygonal metrics. For scale invariant measures, thefourth column indicates whether optimal values are at the top( ↑ ) or bottom ( ↓ ) of the definition range. Polygon measures:inscribed circle (IC), circumscribed circle (CC), polygon area(AR), kernel area (KE), minimum angle (MA), shortest edgelength (SE), and minimum point to point distance (MPD). Metric Abbr. Range Scaleinv. ♯ Edges nE (1 , +∞) – YesInscribed radius IC (0 , ∞) – NoCircumscr. radius CC (0 , ∞) – NoCircle ratio CR [0 , ↑ YesArea AR [0 , ∞) – NoKernel-area KE [0 , ∞) – NoKernal-area ratio KAR [0 , ↑ YesPerimeter-area ratio PAR (0 , ∞) ↑ YesMin. angle MA (0 , 𝜋 ) ↑ YesMax. angle mA (0 , 𝜋 ) ↑ YesShortest edge SE (0 , ∞) – NoEdge ratio ER (0 , ↑ YesMin p2p distance MPD (0 , ∞) – NoNormal. point dist. NPD (0 , ↑ YesShape regularity SRG (0 , ↑ Yes each mesh as a standalone object or to consider each meshas a part of a full data set (when more polygonal meshes areavailable).The former visualization focuses on the polygonal meshas a standalone object. On each available mesh, it visu-ally highlights the element (either triangle or polygon) wherethe minimum and the maximum values are located, togetherwith textual information about them (e.g., the numerical val-ues). This visualization enables the detection of possible ge-ometric degeneracies. The other two visualization approachesconsider the full data set as a single object, and plot how theevaluated geometric metrics evolve in the data set, and howthey correlate among each others.Specifically, a former window shows the variation of theminima and maxima of each geometric metric. Each oneof these linear plots corresponds to a single geometric met-ric, and it is generated by setting its 𝑥 -axis to the indices ofthe meshes in the data set and its 𝑦 -axis the minimum andthe maximum values. The second window is specializedin correlating two user-defined geometric metrics betweeneach other. The correlation is shown by scatter plots, eachof them built by setting its axes to the two selected metrics,respectively (Sect. 3.4). In both windows, plots are inter-active: a single click on a point or line in the plot shows thevalue corresponding to such a point. Also, both windows arehighly customisable by enabling the user setting plot colorsand point/line sizes. Furthermore, plots can be stored on diskas images, and numerical information can be store as textualfiles. PEMesh provides the possibility to solve PEMs on a polyg-onal mesh of the input domain and to visualize the perfor-
Figure 5:
Screenshot of the window visualizing PEM solverresults. On the top, both solver output and ground truth arecolor-mapped on the input polygon meshes, while on the bot-tom a set of linear plots show how solver performances vary inthe data set. mances of any PEM solver. To this end, the PEM solveris not part of the tool, but is rather considered as an exter-nal resource that is invoked from
PEMesh graphical interface.Without loss of generality,
PEMesh assumes that the selectedPEM solver takes an input mesh and returns both the solutionand the ground-truth (if any) of a PDE, together with statis-tics (e.g., approximation error, conditioning of the stiffnessmatrix) on the numerical solvers (Sect. 4.3).When the results of the PEM solver are available, a spe-cialized window graphically shows both the solution and theground-truth on the mesh through a color map, while numer-ical and geometric metrics are represented by customisablelinear plots (Fig. 5). A double click on each plot detachesthe plot itself from the window and enables a full screen vi-sualization. On the right side of the window, a set of visual-ization options provided to customize both color-maps andlinear plots.
PEMesh is intended to be a support for the investigationof possible correlations between geometric and solver per-formance metrics. To reach the goal,
PEMesh provides spe-cialized windows for the graphical visualization via scatterplots of such correlations (Fig. 5). A scatter plot is a plot dis-playing the relationship between two quantitative variablesmeasured on the same input. The values of one variable ap-pear on the horizontal axis, and the values of the other vari-able appear on the vertical axis. This representation allowsus to analyse if the two variables are correlated or not. In theformer case, it describes the direction, form, and strength ofthe relationship. Direction can be either positive (rising) ornegative (falling), while the form can be linear or curvilinear.Finally, the strength is derived from the slope of the plot, andit indicates if the relationship is strong, moderate or weak.
PEMesh provides visualization of possible correlations be-tween two different geometric properties and between a ge-ometric metric and a solver performance evaluation. Theformer is enabled when both geometric metrics and PEMperformance evaluations are available.Given two variables 𝐺 and 𝐺 ′ computed on the same set D. Cabiddu, G. Patanè, M. Spagnuolo:
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EMesh : a Graphical Framework for the Analysis of the Interplay Between Geometry and PEM Solvers(a)(b)
Figure 6:
Scatter plot of the correlation between geometricproperties and performance metrics of the PEM solver: cor-relation between ( 𝑥 -axis) (a) the area-perimeter ratio and (b)the minimum angle with ( 𝑦 -axis) the conditioning number ofthe stiffness matrix of the PEM solver. of input polygon meshes, where 𝐺 is a geometric metric and 𝐺 ′ may either be a geometric metric or a PEM performanceevaluation, PEMesh visualizes their correlation through a spe-cialized scatter plot, with 𝐺 -values on the 𝑥 -axis and 𝐺 ′ -values on the 𝑦 -axis. Similarly to any other plot in the appli-cation, these scatter plots are highly customisable in termsof point color, size, labels, and can be exported as images.An example is shown in Fig. 6.
4. Implementation
PEMesh is a standalone multi-platform desktop applica-tion implemented in C++, and exploits Qt libraries for thedesign and implementation of the graphical user interface(Fig. 1) and Cinolib [12] to generate and visualize meshes.We now discuss the supported data formats (Sect. 4.1), thefamily of parametric polygons available for the generation ofpolygonal meshes (Sect. 4.1), and a description (Sect. 4.3)of the PEM solver used for the experiments on the interplaybetween geometry and analysis.
PEMesh provides the possibility to either load an existingdata set or to generate a new one from scratch. In both cases,
PEMesh supports the most widely used file formats for the ex- (a) 𝑡 = 0 (b) 𝑡 = 1 Figure 7:
Multi-parametric polygon meshes. change of polygonal meshes, namely OBJ, OFF and STL.Furthermore, an additional output format is provided andproduces .node and .ele files encoding vertices and polygonsrespectively. This latter mesh format is provided to supporta large amount of PEM solvers requiring this kind of input.
To generate new polygon meshes from scratch, the useris asked to select one or more polygons to be added to thedomain.
PEMesh provides a list of available polygons of twotypes: parametric and random [2]. When a set of randompolygon is selected, the generated data set is made of a sin-gle mesh; if at least one parametric polygon is chosen, thena family of meshes 𝐷 = { 𝑀 (0) , … , 𝑀 (1)} , is generated,where 𝑀 (0) contains all the parametric polygons at its initialphase (e.g., they do not present critical geometric features(Fig. 7(a)) and they are progressively made worse by a de-formation, controlled by the parameter 𝑡 ∈ [0 , (Fig. 7(b)).In the latter case, the number of generated meshes is user-defined.Other than the polygons in PEMesh ’s list (Fig. 8), the useris allowed to load polygons from file. Such a polygon isautomatically scaled and translated to be placed inside thedomain, and additional editing can be applied by the user it-self.
PEMesh allows us to save the polygon configuration into aCSV file to be possibly reloaded during any following exper-imental session. The CSV stores, for each polygon, any datanecessary to rebuild the configuration (i.e., position, scale,rotation).
As aforementioned,
PEMesh does not include PEM solvers,but is intended to be a support for the analysis of externaltools. Specifically,
PEMesh requires the solver outputs to besaved according to a very simple textual file format. Specifi-cally, both the numerical solution computed by VEM solverand the ground-truth solution (if any) must be as a list oftheir values at the mesh vertices. Finally, these two arraysare saved in a .txt files whose name is composed by the inputfilename and an additional ending to indicate which output itencodes (i.e. either “-solution” or “-ground-truth” respec-tively). For each performance evaluation, an additional .txt file is generated and its name is composed of the input file-name and an additional string to indicate the performance D. Cabiddu, G. Patanè, M. Spagnuolo:
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EMesh : a Graphical Framework for the Analysis of the Interplay Between Geometry and PEM Solvers t = ( r e s t c o n f i g u r a t i o n ) t = . t = . t = . t = . CombConvexityIsotropyMazeN sidedStarU LikeZeta
Figure 8:
The eight families of parametric polygons that gen-erate the meshes in the data set. Each polygon starts froma rest configuration ( 𝑡 = 0 , left column) and progressively de-generates by increasing 𝑡 , thus stressing either one or multiplequality measures in Table 1. name. The single value representing the solver performancemust be written in the file. Use case
As a matter of example, we exploit the
VirtualElement Method (VEM) [26] to demonstrate how the inte-gration between the two tools works. The MATLAB ® codeof the method computes the PDE solution, provides the so-lution ground truth and also computes some evaluations ofPEM solver quality, such as (Fig. 9)• condition number 𝜅 ( 𝐒 ) = ‖ 𝐒 ‖ ‖ 𝐒 −1 ‖ of the PEMstiffness matrix 𝐒 ;• relative error 𝜖 𝑆 ∶= ‖ 𝐮 − 𝐮 𝐡 ‖ 𝑆 ∕ ‖ 𝐮 ‖ 𝑆 , with weightednorm ‖ 𝐯 ‖ 𝑆 = 𝐯 ⊤ 𝐒 𝐯 ;• relative ∞ -error 𝜖 ∞ ∶= ‖ 𝑢 − 𝑢 ℎ ‖ ∞ ∕ ‖ 𝑢 ‖ ∞ , betweenthe ground-truth 𝑢 and the computed 𝑢 ℎ solutions.To enable PEMesh to visualize the results and the correlationbetween PEM solver performances and geometric proper-ties, we wrapper the code into a MATLAB function to becalled from a command line and we redirect the output ofthe
Virtual Element Method to file, according to the file for-mat described in Sec. 4.3. The MATLAB function requiresboth the input mesh and the output directory as parameters.These two simple operations are sufficient to make
PEMesh and the
Virtual Element Method communicate.
5. Discussion and conclusions
We presented
PEMesh , a novel tool helping researchersto perform experimental design and analysis of polygonalmeshes for PEM solvers. Through its easy-to-use graphicalinterface, it simplifies the execution of experimental pipelines,from the design of polygon meshes stressing specific geo-metric properties to the analysis of performances of user-provided PEM solvers. It also allows correlating geometric
Figure 9:
Analysis of the metrics of the numerical solverson parametric families of polygonal meshes of the CHANGEbenchmark. properties with PEM solver performances, and provides ad-vanced visualization modalities of the results such an anal-ysis.
PEMesh is available as an open-source project and weexpect it can be employed in several research activities.
Current limitations and future works
There are severaldirections in which
PEMesh can be improved. First, the cur-rent implementation allows the definition of polygon meshesby loading already exiting polygons, possibly designed byexploiting external tools. Since
PEMesh is intended to sup-port activities in several research fields, additional features,such as the possibility to draw polygons freehand, wouldsimplify the mesh generation process by users coming fromfields other than mesh design. A deeper analysis, includinguser studies, can support the development of an improvedversion of the tool according to user needs.Also,
PEMesh supports 2D meshes, but the entire architec-ture is agnostic to the dimension of the geometric input. It isalmost trivial to extend the graphical user interface to sup-port 3D meshes, but future investigations are necessary on3D mesh generation approaches and definition of geomet-ric properties that would be likely to be of interest for theresearch community.Finally,
PEMesh is designed as a desktop application ex-ploiting RAM memory to both generate meshes and run PEMsolvers. This architecture is sufficient to run preliminary teston sufficiently small meshes, but it is not robust enough tosupport arbitrary large inputs. Future activities will be ad-dressed to improve the underlying architecture and guaranteeefficient independently from the input size.
Acknowledgments
This work is supported by the H2020 ERC AdvancedGrant CHANGE, grant agreement N. 694515. Special thanks
D. Cabiddu, G. Patanè, M. Spagnuolo:
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EMesh : a Graphical Framework for the Analysis of the Interplay Between Geometry and PEM Solvers are given to CNR-IMATI colleagues involved in the CHANGEProject for helpful discussions and comments on polygonalmesh generation and Polyhedral Elements Methods.
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A weak Galerkin finite element method forsecond-order elliptic problems. Journal of Computational and Ap-plied Mathematics 241, 103–115.Daniela Cabiddu is researcher at CNR-IMATI.Since 2013, her research focus is on ComputerGraphics and Geometry Modeling for GIS science,fabrication and engineering. She is also interestedin parallel and distributed computing infrastruc-tures. Her research work has been published oninternational journals and conference proceedings.She has been part of a number of national and Eu-ropean research projects.Giuseppe Patanè is senior researcher at CNR-IMATI. Since 2001, his research is mainly focusedon Computer Graphics. He obtained the NationalScientific Qualification as Full Professor of Com-puter Science. He is author of scientific publica-tions on international journals and conference pro-ceedings, and tutor of Ph.D. and Post.Doc students.He is responsible of R & D activities in national andEuropean projects.Michela Spagnuolo is Research Director at CNR-IMATI and her research interest include geometricand semantic modelling of 3D objects, approaches.On these research topics, she has co-supervised 8PhD thesis (plus one ongoing) and several Lau-rea/Master degree thesis. She authored more than130 reviewed papers in scientific journals and in-ternational conferences, is associate editor of inter-national journals in Computer Graphics. She hasbeen working as scientific responsible of severalinternational and national projects, e.g., REFLEC-TIVE7 Gravitate, H2020 FoF CAxMAN, H2020ErC AdG CHANGE.
D. Cabiddu, G. Patanè, M. Spagnuolo: