MD Dominique Debray
email: dominique.debray@aphp.fr

Harriet Corvol

Chantal Housset

Modifier Genes in Cystic Fibrosis-related Liver Disease

Keywords: ACE, angiotensin I converting enzyme, CF, cystic fibrosis, CFLD, CF-related liver disease, CFTR, cystic fibrosis transmembrane conductance regulator, GSTP1, glutathione cystic fibrosis, cholangiopathy, innate immunity, Serpina1

Introduction

Cystic fibrosis (CF; OMIM 219700) is the most common autosomal recessive disease among

Caucasians. CF is caused by variations in the cystic fibrosis transmembrane conductance regulator (CFTR) gene. Estimates of incidence are of approximately 1 for every 3,000 live births worldwide [START_REF] Rowe | Cystic fibrosis[END_REF] . The secretory defect in CF, leads to a decrease in the luminal hydration of ducts, physicochemical abnormalities of secretions and duct obstruction. It is a multiorgan disease affecting mostly the lungs, pancreas, sweat glands, and the Wolffian ducts in males.

Other clinically relevant manifestations include intestinal obstruction in the neonatal period (meconium ileus) or later in life (distal intestinal obstructive syndrome), and hepatobiliary disease. The median age of survival has increased overtime to exceed 40 years, as a result of neonatal screening and improved management, particularly of lung disease. With the recent development of CFTR-directed therapies that correct the function of the defective protein, life expectancy of individuals with CF should continue to increase [START_REF] Amaral | Novel personalized therapies for cystic fibrosis: treating the basic defect in all patients[END_REF] .

The clinical phenotype in CF shows considerable variability. Both the number of affected organs and the extent of damage within a specific organ vary substantially between patients with CF. This is particularly true as far as the lung and liver disorders are concerned [START_REF] Corvol | Genome-wide association meta-analysis identifies five modifier loci of lung disease severity in cystic fibrosis[END_REF][START_REF] Boelle | including 3,328 patients with CF describes the incidence of CFLD and the impact of ursodeoxycholic acid therapy on the incidence of severe CFLD[END_REF] .

Liver disease has emerged as a significant contributor to the morbidity and mortality of patients with CF. In 2014, CF-related liver disease (CFLD) was the third most common cause of mortality after cardiorespiratory and transplant-related causes, responsible for approximately 2.8% of deaths [START_REF]Cystic Fibrosis Foundation Patient Registry ADRttCD[END_REF] . Focal biliary cirrhosis is the classical histological lesion of CFLD. It has been ascribed to CFTR defect in bile duct epithelial cells (i.e, cholangiocytes), which would result in biliary obstruction by inspissated secretions and progressive periportal fibrosis. Focal biliary cirrhosis has been reported in up to 70% in autopsy studies of CF patients [START_REF] Vawter | Cystic fibrosis in adults: an autopsy study[END_REF] . However, the progression from focal biliary fibrosis to multilobular cirrhosis, which most often manifests as complications of portal hypertension by the end of the second decade of life, occurs in a minority of patients (5 to 10%) [START_REF] Boelle | including 3,328 patients with CF describes the incidence of CFLD and the impact of ursodeoxycholic acid therapy on the incidence of severe CFLD[END_REF][START_REF] Debray | Outcome of cystic fibrosis-associated liver cirrhosis: management of portal hypertension[END_REF][START_REF] Stonebraker | Features of Severe Liver Disease With Portal Hypertension in Patients With Cystic Fibrosis[END_REF][START_REF] Debray | Best practice guidance for the diagnosis and management of cystic fibrosis-associated liver disease[END_REF][START_REF] Colombo | Liver disease in cystic fibrosis: A prospective study on incidence, risk factors, and outcome[END_REF][START_REF] Lindblad | Natural history of liver disease in cystic fibrosis[END_REF][START_REF] Lamireau | Epidemiology of liver disease in cystic fibrosis: a longitudinal study[END_REF][START_REF] Debray | Cystic Fibrosis-related Liver Disease: Research Challenges and Future Perspectives[END_REF] . In the largest series of CF patients with cirrhosis and portal hypertension (561 in total), 90% were diagnosed by 18 years of age with a mean age at diagnosis of 10 years [START_REF] Stonebraker | Features of Severe Liver Disease With Portal Hypertension in Patients With Cystic Fibrosis[END_REF] . The progression to cirrhosis and portal hypertension remains unpredictable in the absence of sensitive biomarkers [START_REF] Debray | Cystic Fibrosis-related Liver Disease: Research Challenges and Future Perspectives[END_REF] . The other biliary manifestations of CF, including large-duct sclerosing cholangitis, microgallbadder, gallbladder dyskinesia, and symptomatic cholelithiasis have been reported with a wide range of prevalence [START_REF] Assis | Gallbladder and bile duct disease in Cystic Fibrosis[END_REF] . The factors responsible for this phenotypic variability are not known. Moreover, the pathophysiology of liver disease remains uncertain and so far, no therapy has proven effective to prevent the progression of CFLD [START_REF] Boelle | including 3,328 patients with CF describes the incidence of CFLD and the impact of ursodeoxycholic acid therapy on the incidence of severe CFLD[END_REF][START_REF] Debray | Cystic Fibrosis-related Liver Disease: Research Challenges and Future Perspectives[END_REF][START_REF] Cheng | Ursodeoxycholic acid for cystic fibrosis-related liver disease[END_REF] . A full understanding of the pathophysiology and the effect of environmental and non-CFTR genetic influences in the context of CFLD development would help improve management and develop new drug therapies.

This review provides the latest knowledge in the pathophysiology of CFLD and CF genetic modifier research in CFLD.

Pathophysiology of CFLD: what is known?

CFTR expression in the hepatobiliary system, is restricted to the apical membrane of cholangiocytes lining the intra-and extrahepatic bile ducts. CFTR drives fluid and bicarbonate secretion in cholangiocytes [START_REF] Cohn | Localization of the cystic fibrosis transmembrane conductance regulator in human bile duct epithelial cells[END_REF][START_REF] Chinet | Regulation of electrogenic anion secretion in normal and cystic fibrosis gallbladder mucosa[END_REF] . It has been assumed that the importance of CFTR in maintaining homeostasis in the biliary tree was related to the "bicarbonate umbrella" hypothesis, in which a proper alkaline balance is critical to prevent cholangiocyte injury by hydrophobic bile acids [START_REF] Beuers | The biliary HCO(3)(-) umbrella: a unifying hypothesis on pathogenetic and therapeutic aspects of fibrosing cholangiopathies[END_REF][START_REF] Beuers | The biliary HCO(3)(-) umbrella: experimental evidence revisited[END_REF] . According to this hypothesis, the loss of CFTR function would contribute to pH dysregulation and thereby would promote cholangiocyte injury [START_REF] Debray | Cystic Fibrosis-related Liver Disease: Research Challenges and Future Perspectives[END_REF] . Evidence has also been provided to indicate that the loss of CFTR affected innate immunity and promoted gut dysbiosis, which in turn would induce intestinal inflammation, endotoxinemia and CFLD 20-24 . New functions of CFTR in cholangiocytes were recently described, i.e., the regulation of cell polarity, and of toll-like receptor 4 (TLR4) response via the activation of the tyrosine kinase Src [START_REF] Fiorotto | The cystic fibrosis transmembrane conductance regulator controls biliary epithelial inflammation and permeability by regulating Src tyrosine kinase activity[END_REF] . This latter study showed that in cholangiocytes, CFTR promoted the assembly of a protein complex that maintained Src in an inactive state [START_REF] Fiorotto | The cystic fibrosis transmembrane conductance regulator controls biliary epithelial inflammation and permeability by regulating Src tyrosine kinase activity[END_REF] . Thus, in the biliary epithelium exposed to endotoxins, when CFTR is defective, this complex does not assemble, resulting in the self-activation of Src, and subsequent activation of TLR4 and of nuclear factor kappa-lightchain-enhancer of activated B cells (NF-kB). This mechanism may contribute to an increase in the production of proinflammatory cytokines leading to peribiliary inflammation and fibrosis [START_REF] Fiorotto | The cystic fibrosis transmembrane conductance regulator controls biliary epithelial inflammation and permeability by regulating Src tyrosine kinase activity[END_REF] . Moreover, the impairment in barrier function is expected to increase the back-diffusion of toxic bile acids, and thereby to further aggravate peribiliary inflammation and fibrogenesis. We alo recently showed in CF mouse models, that the development of CFLD features required the combination of CFTR defect with both genetic and environmental factors such as diet-induced dysbiosis [START_REF] Debray | Diet-induced dysbiosis and genetic background synergize with cystic fibrosis transmembrane conductance regulator deficiency to promote cholangiopathy in mice[END_REF][START_REF] Debray | Defects in gallbladder emptying and bile Acid homeostasis in mice with cystic fibrosis transmembrane conductance regulator deficiencies[END_REF] .

Molecular mechanisms underlying phenotypic differences in CFLD

As an approach to unravel the molecular mechanisms underlying phenotypic differences in CFLD, investigators sought to determine if the presence and/or severity of CFLD was associated with a transcriptomic signature. Pereira et al. provided evidence for a transcriptional basis of CFLD pathogenesis [START_REF] Pereira | Transcriptional basis for hepatic fibrosis in cystic fibrosis-associated liver disease[END_REF] . The liver biopsies of children with CFLD compared with those of children with biliary atresia and normal controls displayed differences in the expression of multiple genes related to tissue remodeling and fibrogenesis including chemokines, collagens and matrix metalloproteases. In CFLD, regardless of fibrosis severity, compared with normal controls and biliary atresia, the expression of genes involved in tissue remodeling was decreased, notably the expression of plasminogen activator inhibitor-1 (PAI-1, up to 25-fold) and of tissue inhibitor of metalloproteinase-1 (TIMP-1). By contrast, the expression of genes encoding matrix metalloproteases or involved in the immue response such as interleukins and their receptors were overexpressed in the liver of patients with CFLD [START_REF] Pereira | Transcriptional basis for hepatic fibrosis in cystic fibrosis-associated liver disease[END_REF] . PAI-1 and TIMP-1 are two key regulators of matrix remodeling. PAI-1, encoded by the gene SERPINE1, belongs to a family of serine protease inhibitors. It acts as an inhibitor of the tissue-type and urokinase-type plasminogen activators, which convert plasminogen into plasmin [START_REF] Kruithof | Plasminogen activator inhibitors--a review[END_REF] , a protease required for the activation of matrix metalloproteases and soluble factors such as hepatocyte growth factor or transforming growth factor b. TIMP-1 is an inhibitor of matrix metalloproteases. Therefore, the decrease in the production of PAI-1 and TIMP-1 is expected to increase the activation of matrix metalloproteases and ultimately to increase matrix turnover. Of particular interest, another member of the same family of serine protease inhibitors, namely SERPINA1, has been identified as a modifier gene of CFLD [START_REF] Bartlett | Genetic modifiers of liver disease in cystic fibrosis[END_REF] . Because the genes encoding PAI-1 and CFTR are both located on the long arm of chromosome 7, future studies should also determine whether there is any genetic linkage between these two loci. Overall, transcriptomic analyses supported the role of cytokine and chemokine signaling pathways, hepatic stellate cell activation and matrix remodeling in the pathogenesis of CFLD, as suggested by several studies [START_REF] Ramm | Fibrogenesis in pediatric cholestatic liver disease: role of taurocholate and hepatocyte-derived monocyte chemotaxis protein-1 in hepatic stellate cell recruitment[END_REF][START_REF] Lewindon | The role of hepatic stellate cells and transforming growth factor-beta(1) in cystic fibrosis liver disease[END_REF][START_REF] Pereira | Serum markers of hepatic fibrogenesis in cystic fibrosis liver disease[END_REF] .

We recently demonstrated that CFLD, i.e., cholangiopathy features, was markedly influenced by the genetic background in Cftr knockout mice [START_REF] Debray | Diet-induced dysbiosis and genetic background synergize with cystic fibrosis transmembrane conductance regulator deficiency to promote cholangiopathy in mice[END_REF] . By comparing Cftr knockout mice in a congenic C57BL/6J or mixed C57BL/6J;129/Ola background, we found that cholangiopathy occurred only in mice of the congenic background. Transcriptomic analyses of the liver showed that mice in the congenic versus mixed background, overexpressed genes conveying an hyperinflammatory state whereas genes related to innate and adaptative immunity were underexpressed, including the absence of CD1d2 expression, in the C57BL/6J background.

Protein-protein interactions predicted by the analysis of genes overexpressed in the liver of the congenic mice formed a large network, displaying links between inflammation, fibrosis and cell (presumably cholangiocyte) proliferation [START_REF] Debray | Diet-induced dysbiosis and genetic background synergize with cystic fibrosis transmembrane conductance regulator deficiency to promote cholangiopathy in mice[END_REF] . This result supported the view that cholangiopathy arising in CF, is the result of an ill-adapted innate immune response to endotoxins coming from the intestine and triggering a pro-inflammatory response [START_REF] Fiorotto | The cystic fibrosis transmembrane conductance regulator controls biliary epithelial inflammation and permeability by regulating Src tyrosine kinase activity[END_REF] .

Phenotype-genotype correlations

Despite all efforts provided by investigators to elucidate the pathophysiology of CFLD, the reason why only one-third of patients with CF develop liver disease, and why the severity of CFLD is so highly variable remains poorly known.

There are currently more than 2,000 variations listed in the CFTR mutation databases (https://cftr2.org). Among all CFTR variations, the most prevalent one, accounting for approximately 70% of all CFTR alleles worldwide, results in the deletion of a phenylalanine at position 508 (F508del). CFTR variations are categorized into six classes based on their functional consequences and subsequent residual activity at the apical membrane of epithelial cells. Class I, II, and III mutations (i.e., severe mutations) result in a complete loss of the chloride-channel function; in contrast, class IV, V, and VI mutations (i.e., mild mutations) are associated with altered conductance properties, reduced synthesis or defective stability of normal CFTR, compatible with residual CFTR membrane activity [START_REF] Rowe | Cystic fibrosis[END_REF] .

CFLD is largely limited to pancreatic insufficient patients with biallelic loss-of-function mutations in CFTR (i.e., class I, II, or III mutations on both allele) [START_REF] Boelle | including 3,328 patients with CF describes the incidence of CFLD and the impact of ursodeoxycholic acid therapy on the incidence of severe CFLD[END_REF][START_REF] Colombo | Liver disease in cystic fibrosis: A prospective study on incidence, risk factors, and outcome[END_REF][START_REF] Lamireau | Epidemiology of liver disease in cystic fibrosis: a longitudinal study[END_REF][START_REF] Colombo | Analysis of risk factors for the development of liver disease associated with cystic fibrosis[END_REF][START_REF] Duthie | Genotype analysis for delta F508, G551D and R553X mutations in children and young adults with cystic fibrosis with and without chronic liver disease[END_REF] . Meconium ileus was also shown to be a risk factor for severe CF liver disease with portal hypertension [START_REF] Boelle | including 3,328 patients with CF describes the incidence of CFLD and the impact of ursodeoxycholic acid therapy on the incidence of severe CFLD[END_REF][START_REF] Debray | Outcome of cystic fibrosis-associated liver cirrhosis: management of portal hypertension[END_REF][START_REF] Colombo | Liver disease in cystic fibrosis: A prospective study on incidence, risk factors, and outcome[END_REF][START_REF] Lamireau | Epidemiology of liver disease in cystic fibrosis: a longitudinal study[END_REF][START_REF] Maurage | Meconium ileus and its equivalent as a risk factor for the development of cirrhosis: an autopsy study in cystic fibrosis[END_REF][START_REF] Chryssostalis | Liver disease in adult patients with cystic fibrosis: a frequent and independent prognostic factor associated with death or lung transplantation[END_REF] . There is also an increased prevalence of severe CFLD in males (2:1) over females, in keeping with the fact that male gender is a factor of fibrosis progression in all chronic liver diseases [START_REF] Boelle | including 3,328 patients with CF describes the incidence of CFLD and the impact of ursodeoxycholic acid therapy on the incidence of severe CFLD[END_REF][START_REF] Stonebraker | Features of Severe Liver Disease With Portal Hypertension in Patients With Cystic Fibrosis[END_REF] .

Recently, we have confirmed in a large cohort of CF patients that the F508del homozygous CFTR genotype was a risk factor for the development of liver involvement and severe CFLD [START_REF] Boelle | including 3,328 patients with CF describes the incidence of CFLD and the impact of ursodeoxycholic acid therapy on the incidence of severe CFLD[END_REF] .

However, the heterogeneous phenotype of liver involvement in CF patients bearing the same CFTR genotype, including CF siblings, suggests that non-CFTR genetic variations, i.e., modifier genes contribute to the risk of severe CFLD [START_REF] Gallati | Disease-modifying genes and monogenic disorders: experience in cystic fibrosis[END_REF][START_REF] Salvatore | Genotype-phenotype correlation in cystic fibrosis: the role of modifier genes[END_REF] , and to the liver phenotypic heterogeneity, as shown for lung and intestinal phenotypes in CF [START_REF] Corvol | Genome-wide association meta-analysis identifies five modifier loci of lung disease severity in cystic fibrosis[END_REF][START_REF] Drumm | Genetic modifiers of lung disease in cystic fibrosis[END_REF][START_REF] Blackman | Relative contribution of genetic and nongenetic modifiers to intestinal obstruction in cystic fibrosis[END_REF] . A poor concordance of liver disease in sibling pairs with CF has been reported, which argues against a major role of environmental factors [START_REF] Castaldo | Liver expression in cystic fibrosis could be modulated by genetic factors different from the cystic fibrosis transmembrane regulator genotype[END_REF] .

The CF genetic modifier research

Initial searches for non-CFTR genetic variants that might influence liver phenotype in patients with CF, used a candidate gene approach, which is based on biological plausibility. However, this approach has not been conclusive so far, mainly because the understanding of the pathophysiology of CF liver disease was insufficient. Genes that have been studied as putative modifiers of CF liver disease include SERPINA1, encoding the alpha-1-antitrypsin protein, genes which variants cause hemochromatosis (HFE, TFR2 and FPN1), and genes encoding glutathione S-transferase P1 (GSTP1), mannose binding lectin 2 (MBL2), angiotensin I converting enzyme (ACE), transforming growth factor beta 1 (TGFB1), interleukin 8 (currently known as CXCL8), as well as ATP-binding cassette subfamily B member 4, which is defective in progressive familial intrahepatic cholestasis type 3, intrahepatic cholestasis of pregnancy and drug-induced cholestasis [START_REF] Bartlett | Genetic modifiers of liver disease in cystic fibrosis[END_REF][START_REF] Henrion-Caude | Liver disease in pediatric patients with cystic fibrosis is associated with glutathione S-transferase P1 polymorphism[END_REF][START_REF] Gabolde | The mannose binding lectin gene influences the severity of chronic liver disease in cystic fibrosis[END_REF][START_REF] Tomaiuolo | An MBL2 haplotype and ABCB4 variants modulate the risk of liver disease in cystic fibrosis patients: a multicentre study[END_REF][START_REF] Arkwright | End-organ dysfunction in cystic fibrosis: association with angiotensin I converting enzyme and cytokine gene polymorphisms[END_REF] . However, only SERPINA1 has been confirmed as a modifier of liver disease severity in CF. Most studies have investigated cohorts of small size and phenotyping did not address the development of severe (biliary) cirrhosis associated with portal hypertension. The largest two-stage control study of CFLD (defined as advanced liver disease with portal hypertension) focused on 9 variations in 5 candidate genes (SERPINA1, ACE, GSTP1, MBL2, and TGFB1) that had previously been screened in patients with various degree of CFLD severity [START_REF] Bartlett | Genetic modifiers of liver disease in cystic fibrosis[END_REF][START_REF] Henrion-Caude | Liver disease in pediatric patients with cystic fibrosis is associated with glutathione S-transferase P1 polymorphism[END_REF][START_REF] Gabolde | The mannose binding lectin gene influences the severity of chronic liver disease in cystic fibrosis[END_REF][START_REF] Arkwright | End-organ dysfunction in cystic fibrosis: association with angiotensin I converting enzyme and cytokine gene polymorphisms[END_REF] . Genetic analyses of the initial cohort (124 patients with CFLD and 843 control patients without CFLD) showed that a single copy of the SERPINA1 Z allele and each additional copy of the TGFB1 codon 10 C allele were associated with significantly increased odds of CFLD [START_REF] Bartlett | Genetic modifiers of liver disease in cystic fibrosis[END_REF] . In the replication study including an additional 136 patients with CFLD and 1,088 without CFLD, the SERPINA1 Z allele was confirmed as a modifier of liver disease in CF, but not the TGFB1 codon 10 variant. Overall, this unique two-stage control study showed a strong association of the Z-allele of the a 1 -anti-protease (SERPINA1) gene with CFLD in a total of 260 patients with CFLD versus 1,931 patients with CF and no CFLD. Of particular interest with respect to pathophysiology, the alpha-1-antitrypsin variant protein accumulates not only in hepatocytes but also in cholangiocytes, in subjects bearing the SERPINA1 Z alleles [START_REF] Callea | Storage of alpha-1-antitrypsin in intrahepatic bile duct cells in alpha-1-antitrypsin deficiency (Pi Z phenotype)[END_REF] . One may speculate that the accumulation of alpha-1-antitrypsin variant induces an endoplasmic reticulum stress in cholangiocytes as it does in hepatocytes.

This may aggravate endoplasmic reticulum stress induced by the accumulation of F508del CFTR and predispose cholangiocytes to injury of other causes, e.g. endotoxins or toxic bile.

While genome-wide association data have not yet been reported in CFLD, a different methodology was used in a recent work using a system biology approach combining database and literature mining, gene expression study, network analysis, pathway enrichment analysis and protein-protein interactions to examine functional relationships between reported genes [START_REF] Trouve | In silico search for modifier genes associated with pancreatic and liver disease in Cystic Fibrosis[END_REF] . This large study led to the identification of novel modifier genes potentially involved in CFLD: SLC33A1 (NCBI ID: 9197), GPNMB (NCBI ID: 10457), NCF2 (NCBI ID: 4688), RASGRP1 (NCBI ID: 10125), LGALS3 (NCBI ID: 3958) and PTPN13 (NCBI ID: 5783) [START_REF] Trouve | In silico search for modifier genes associated with pancreatic and liver disease in Cystic Fibrosis[END_REF] . Pathways associated with these genes highlighted the innate immune system and Ubiquitin C (UBC, OMIM: 191340) as the central node linking CF to liver disease [START_REF] Trouve | In silico search for modifier genes associated with pancreatic and liver disease in Cystic Fibrosis[END_REF][START_REF] Cui | Mechanisms and pathways of innate immune activation and regulation in health and cancer[END_REF] .

In conclusion, CFLD is characterized by progressive damage of bile ducts with inflammation, and peribiliary fibrosis in portal tracts. There is increasing evidence to indicate that CFTR deficiency in the biliary epithelium induces an increased vulnerability to injury of different causes including bacterial endotoxin and a hyperinflammatory response of cholangiocytes or in their vicinity. The reason why, among CF patients with the same CFTR genoytype, a minority progress to multilobular cirrhosis in the first decade of life still remains poorly understood but genes involved in the innate immune system are candidate modifiers. A multifaceted approach combining genome-wide association studies, and gene expression studies will hopefully reveal significant gene modifiers for CF liver disease. Challenges for the future are to develop novel therapies to prevent or halt the progression of this disease.

Key points

• Cystic fibrosis (CF)-related liver disease (CFLD) manifests as focal biliary fibrosis in approximately one third of patients, but fibrosis may progress to multilobular cirrhosis by the end of the second decade of life in a minority of patients (5 to 10%).

• The heterogeneous severity of liver involvement in patients bearing the same CFTR genotype, including CF siblings, suggests that non-CFTR genetic variations contribute to the risk of severe liver disease • So far, the only modifier gene validated in CFLD is SERPINA1 (alpha-1-antitrypsin) Z allele • Evidence has recently been provided to indicate that cholangiopathy arising in CF, is the result of an ill-adapted innate immune response to endotoxins coming from the intestine and triggering a pro-inflammatory response • Using a system biology approach, novel modifier genes potentially involved in CFLD were identified and associated pathways highlighted the innate immune system and Ubiquitin C (UBC, OMIM: 191340), as the central node linking CF to liver disease.

• Genome-wide association data, not yet available, are urgently needed to identify potential targets for modulating liver disease severity in CF.

References and recommended reading

Papers of particular interest, published within the annual period of review, have been

Financial support and sponsorship: This work was supported by the Cystic Fibrosis Patients Association Vaincre la Mucoviscidose, Paris, France


Chady Ghnatios
email: cghnatios@ndu.edu.lb

Emmanuelle Abisset
email: emmanuelle.abisset-chavanne@ensam.eu

Amine Ammar
email: amine.ammar@ensam.eu

Elías Cueto
email: ecueto@unizar.es

Jean-Louis Louis Duval
email: jean-louis.duval@esi-group.com

Francisco Chinesta
email: francisco.chinesta@ensam.eu

Advanced separated spatial representations for hardly separable domains

Keywords: PGD, Space separated representation, Parametric modeling

come

Introduction

When looking for an approximation of the solution u(x, t) of a given PDE, here assumed, without loss of generality, to be scalar and linear, the standard finite element method considers the approximation

u(x, t) = N ∑ i=1 U i (t)N i (x), (1)
where U i denotes the value of the unknown field at node X i and N i (x) represents the so-called shape function associated to the ith node. Here N refers to the number of nodes employed to approximate the field u in the domain Ω in which the physical problem is defined. This approximation results in an algebraic problem of size N in the linear case, or the iterative solution of many of them in the general transient and nonlinear case. In order to alleviate the computational cost, model order reduction techniques were proposed and are nowadays intensively used.

In the framework of POD-based model order reduction [START_REF] Chinesta | Model Order Reduction[END_REF], a learning stage allows extracting the significant modes φ i (x) that best approximate the solution. In many cases, a reduced number of modes R (R ≪ N) suffice to approximate the solution of problems similar to the one that served to extract the modes in the learning stage.

Thus, the solution u(x, t) is projected onto the reduced basis composed of functions {φ 1 (x), . . . , φ R (x)}, according to

u(x, t) ≈ R ∑ i=1 ξ i (t)φ i (x), (2)
that now requires the solution of linear systems of size R instead the ones of size N characteristic of finite element solutions. This often allows us to obtain impressive computing time savings. Approximations (1) or (2) imply a finite sum of time-dependent coefficients and space functions. The last are assumed known: they consist of the usual finite element shape functions or the modes extracted by applying the POD. A generalization of this procedure could consist on assuming that space functions are also unknown and therefore to compute both, time and space functions, on the fly [START_REF] Ladevèze | The large time increment method for the analyze of structures with nonlinear constitutive relation described by internal variables[END_REF]. Thus, this approximation reads

u(x, t) ≈ M ∑ i=1 T i (t)X i (x). (3)
Since both functions involved in approximation (3) are unknown, the problem solution becomes nonlinear, and its solution requires an appropriate linearization strategy. The interested reader can refer to [START_REF] Chinesta | The Proper Generalized Decomposition for Advanced Numerical Simulations. a Primer[END_REF][START_REF] Cueto | Proper Generalized Decompositions: An Introduction to Computer Implementation with Matlab[END_REF][START_REF] Pierre | The latin multiscale computational method and the proper generalized decomposition[END_REF][START_REF] Azaiez | A new algorithm of proper generalized decomposition for parametric symmetric elliptic problems[END_REF][START_REF] Perotto | Model reduction by separation of variables: a comparison between Hierarchical Model reduction and Proper Generalized Decomposition[END_REF][START_REF] Reyes | Reduced order models for thermally coupled low mach flows[END_REF][START_REF] Senecal | Characterization of the proper generalized decomposition method for fixed-source diffusion problems[END_REF] and the references therein for practical details on the computer implementation of separated representations.

Expression (3) evidences that the solution procedure requires the solution of about M problems, with M ≪ N and M ∼ R (slightly more, in fact, because of the nonlinearity induced by separated representations) involving the space coordinates (in general three-dimensional and whose associated discrete systems are of size N) to compute the space functions X i (x) and about M one-dimensional problems to calculate the time functions T i (t). Due to the fact that the computing cost related to the solution of 1D problems is negligible with respect to the solution of 3D problems, the resulting computational complexity reduces drastically, scaling with M instead of P (here, P represents the number of time steps considered in the time domain discretization and that corresponds with the complexity of standard incremental time-integration techniques).

Another step forward consisted in assuming model parameters as extra-coordinates. Thus, space-time-parameters separated representations allowed constructing the so-called computational vademecums (also known as abacus, virtual charts, nomograms, . . . ) efficiently employed for multiple purposes: simulation, optimization, inverse analysis, uncertainty propagation and simulation-based control, all of them under real-time constraints [START_REF] Chinesta | Model Order Reduction[END_REF][START_REF] Chinesta | PGD-Based computational vademecum for efficient design, optimization and control[END_REF]. When the unknown field involves space, time and a series of parameters µ 1 , . . . , µ Q , its associated separated representation reads

u(x, t, µ 1 , . . . , µ Q ) ≈ M ∑ i=1 X i (x)T i (t) Q ∏ j=1 M j i (µ j ).
(4)

Space separation

The separation of space coordinates was also intensively considered in our former works referred later, as well as in other references, see, for instance, [START_REF] Courard | Integration of PGD-virtual charts into an engineering design process[END_REF] and the references therein. Space separation allowed addressing multi-physics problems defined in degenerated geometries in which at least one of its dimensions results to be much smaller that the other ones (e.g. beams, plates, shells, laminates) or processes involving additive layers (e.g., automated tape placement, 3D printing, or additive manufacturing). Thus, if domain Ω can be decomposed as Ω = Ω x × Ω y × Ω z , the solution u(x, y, z) could be approximated by using the separated representation

u(x, y, z) ≈ M ∑ i=1 X i (x)Y i (y)Z i (z), (5)
that allows calculating the 3D solution from a sequence of 1D problems. For some geometries, as the ones concerning plates or shells, in-plane-out-of-plane separated representations become specially appealing,

u(x, y, z) ≈ M ∑ i=1 X i (x, y)Z i (z), (6)
where the 3D complexity is reduced to a 2D complexity, related to the calculation of in-plane functions X i (x, y).

As discussed in [START_REF] Chinesta | Model Order Reduction[END_REF], the cost savings provided by the use of these separated representations are potentially impressive when the spatial domain is fully separable. Indeed, the complexity of the simulation now scales with the one-dimensional meshes used to solve the BVP's in Ω x , Ω y and Ω z , associated to the computation of functions X i (x), Y i (y) and Z i (z) or with the two-dimensional ones associated with the calculation of functions X i (x, y) in the case of in-plane-out-of-plane separated representations [START_REF] Chinesta | The Proper Generalized Decomposition for Advanced Numerical Simulations. a Primer[END_REF].

When the domain is not intrinsically separable or, in other words, expressible from a direct cartesian product, fully separated representations require the use of appropriate geometrical mappings [START_REF] Courard | Integration of PGD-virtual charts into an engineering design process[END_REF][START_REF] Ammar | Parametric solutions involving geometry: a step towards efficient shape optimization[END_REF] or the immersion of the non-separable domain onto a fully separable one [START_REF] Gonzalez | Recent advances in the use of separated representations[END_REF][START_REF] Ch | On the space separated representation when addressing the solution of PDE in complex domains[END_REF].

In-plane-out-of-plane separated representations are particularly useful for addressing the solution of problems defined in plate and shell geometries, [START_REF] Bognet | Advanced simulation of models defined in plate geometries: 3D solutions with 2D computational complexity[END_REF] and [START_REF] Bognet | Separated representations of 3D elastic solutions in shell geometries[END_REF] respectively, or extruded domains [START_REF] Leygue | Towards a framework for non-linear thermal models in shell domains[END_REF]. A parametric 3D elastic solution of beams involved in frame structures was proposed in [START_REF] Bordeu | Parametric 3D elastic solutions of beams involved in frame structures[END_REF]. The same approach was extensively considered in structural plate and shell models in [START_REF] Gallimard | Coupling finite element and reliability analysis through proper generalized decomposition model reduction[END_REF][START_REF] Vidal | Composite beam finite element based on the proper generalized decomposition[END_REF][START_REF] Vidal | Proper generalized decomposition and layer-wise approach for the modeling of composite plate structures[END_REF][START_REF] Vidal | Explicit solutions for the modeling of laminated composite plates with arbitrary stacking sequences[END_REF][START_REF] Vidal | Shell finite element based on the proper generalized decomposition for the modeling of cylindrical composite structures[END_REF][START_REF] Vidal | Assessment of variable separation for finite element modeling of free edge effect for composite plates[END_REF] and [START_REF] Pruliere | 3D Simulation of laminated shell structures using the proper generalized decomposition[END_REF].

Space separated representations where enriched with discontinuous functions for representing cracks in [START_REF] Giner | The proper generalized decomposition (PGD) as a numerical procedure to solve 3D cracked plates in linear elastic fracture mechanics[END_REF], delamination [START_REF] Metoui | The proper generalized decomposition for the simulation of delamination using cohesive zone model[END_REF] and thermal contact resistances in [START_REF] Chinesta | First steps towards an advanced simulation of composites manufacturing by automated tape placement[END_REF]. The in-plane-out-of-plane decomposition was then extended to many other physics like squeeze flows of Newtonian and non-Newtonian fluids in layered domains in [START_REF] Ch | The squeeze flow of composite laminates[END_REF] or electromagnetism [START_REF] Tertrais | On the proper generalized decomposition applied to microwave processes involving multilayered components[END_REF].

Paper outline

The present paper aims to propose a new simple procedure for treating, in a fully separated representation manner, problems in which the expression of the parametric dependency in a separated form was considered difficult, or where the separated representation performed by invoking SVD-like techniques had a significant impact in the performance and efficiency of the resulting numerical procedures.

After the just addressed introduction on separated representations, with special emphasis in those involving space coordinates, the next section will introduce the main idea of the present work. Namely, a very efficient geometrical mapping leading to a separable description of the problem. Then, in Section 3, the proposed approach is applied to the solution of thermal problems defined in thin domains, but in which the material conductivity has not a compact separated representation. Section 4 addresses the issue of 1D thermal problems containing an inclusion of a different thermal conductivity and whose position in the domain is assumed to be the model parameter. Consequently, the temperature field for any position of the inclusion is found. The paper finishes by summarizing the most valuable conclusions.

Domain mapping

We introduce the proposed methodology, for the sake of simplicity and without loss of generality, by making use of a two-dimensional heat transfer problem. Thus, we consider the temperature field u(x), x = (x, y), defined in a thin domain Ω = (0, L) × (0, H ), with H ≪ L, where a non-planar internal boundary (as the one sketched in Fig. 1) separates the upper domain Ω u , with thermal conductivity K u , and the bottom domain, Ω b , with conductivity K b . This boundary is defined by a function h(x). The temperature field is assumed to be the solution of the steady-state heat transfer problem defined in Ω , whose weak form reads

∫ Ω K (x)∇u * • ∇u d x = 0, (7)
with u ∈ H 1 (Ω ) space and u * ∈ H 1 0 (Ω ). The problem is subjected to the boundary conditions

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂u(x, y) ∂ x ⏐ ⏐ ⏐ ⏐ x=0 = 0, ∂u(x, y) ∂ x ⏐ ⏐ ⏐ ⏐ x=L = 0, u(x, 0) = 0, u(x, H ) = u g , (8)
where u g denotes the prescribed, temperature on the top surface.

Because of the small domain thickness, i.e., H ≪ L, standard mesh-based discretization techniques fail due to the excessive number of degrees of freedom due to the domain degeneracy, i.e. the very different characteristic dimensions, with the thickness dimension orders of magnitude smaller that the others. Those based on the use of separated representations

u(x, y) ≈ M ∑ i=1 X i (x)Y i (y), (9)
are faced to the difficulties relative to the presence of the non-planar boundary, that implies a hardly separable conductivity field, that as discussed below needs too many modes, M, to converge, that is, for reaching a residual small enough [START_REF] Chinesta | The Proper Generalized Decomposition for Advanced Numerical Simulations. a Primer[END_REF]. Of course, conductivity can always be separated by invoking, for instance, the singular value decomposition (SVD), i.e., by expressing the conductivity as

K (x, y) ≈ K ∑ k=1 F k (x)G k (y). ( 10
)
In the case of a planar boundary Γ = Ω u ∩ Ω b , characterized by h(x) = h, with 0 < h < H , a single term suffices for separating the conductivity, i.e.,

K p (x, y) = F p (x)G p (y), (11)
where the superscript (•) p refers to the planar interface configuration, F p (x) = 1 and G p (x) is defined by

G p (y) = K u -(K u -K b )χ(y), (12)
with

χ(y) = { 1 y < h 0 y ≥ h . ( 13
)
However, when internal boundaries deviate from the planar configuration, the number of modes K increases prohibitively, and with it the operators involved in the weak form [START_REF] Perotto | Model reduction by separation of variables: a comparison between Hierarchical Model reduction and Proper Generalized Decomposition[END_REF]. Thus, both the performance and the efficiency of the solver are seriously compromised.

In order to circumvent this issue, we define two mappings transforming Ω u and Ω b into two rectangular domains, noted respectively R u and R b .

1. Mapping Ω b into R b . The first mapping with r = (r, s) ∈ R b = (0, L) × (0, 1) reads { x = r, y = s h(r ), (14)
or, equivalently,

{ r = x, s = y h(x) . (15)
The components of the Jacobian matrix J b result

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ x ∂r = 1, ∂ x ∂s = 0, ∂ y ∂r = s h ′ (r ), ∂ y ∂s = h(r ), (16)
that allows transforming the differential operators involved in the weak form [START_REF] Perotto | Model reduction by separation of variables: a comparison between Hierarchical Model reduction and Proper Generalized Decomposition[END_REF],

∇ = ( ∂ ∂ x , ∂ ∂ y ) T as ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ∂• ∂ x = ∂• ∂r ∂r ∂ x + ∂• ∂s ∂s ∂ x = ∂• ∂r - ∂• ∂s s h ′ (r ) h(r ) , ∂• ∂ y = ∂• ∂r ∂r ∂ y + ∂• ∂s ∂s ∂ y = ∂• ∂s 1 h(r ) , (17)
that, together with

d x = det(J b )dr = h(r )dr, (18)
allows rewriting the weak form [START_REF] Perotto | Model reduction by separation of variables: a comparison between Hierarchical Model reduction and Proper Generalized Decomposition[END_REF]. For this purpose, we define matrix

B b B b = ⎛ ⎜ ⎜ ⎝ 1 -s h ′ (r ) h(r ) 0 1 h(r ) ⎞ ⎟ ⎟ ⎠ , (19)
and the gradient operator ∇ r = ( ∂ ∂r , ∂ ∂s ) T , such that

∇• = B b ∇ r •, (20)
that allows writing the weak form as

∫ R b K b ( ∇ r u * ) T B T b B b ∇ r u det(J b )dr = 0. ( 21
)
2.

Mapping Ω u into R u . Equivalently, for Ω u we define

{ x = r, y = (s -1)(H -h(r )) + h(r ), (22)
with (r, s) ∈ R u = (0, L) × (1, 2).

The gradient operator and the Jacobian are then calculated for this mapping following the same rationale considered in the previous approach. The final weak form in R u reads ∫

R u K u ( ∇ r u * ) T B T u B u ∇ r u det(J u )dr = 0, (23)
where B u is now given by

B u = ⎛ ⎜ ⎜ ⎝ 1 h ′ (s -2) H -h 0 1 H -h ⎞ ⎟ ⎟ ⎠ . ( 24
)
By defining the characteristic functions of both subdomains, χ u (r) and χ b (r),

χ u (r) = { 1 if s ≥ 1, 0 if s < 1, (25)
and

χ b (r) = { 1 if s < 1, 0 if s ≥ 1. ( 26
)
By using the notations

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ K (r) = χ u (r)K u + χ b (r)K b , B(r) = χ u (r)B u (r) + χ b (r)B b (r), det(J(r)) = χ u (r) det(J u (r)) + χ b (r) det(J b (r)), (27)
the weak form in

R = R u ∪ R b reads ∫ R K (r) ( ∇ r u * ) T B T (r)B(r)∇ r u det(J(r)) dr = 0. (28)
Now, u(r, s) is approximated in the separated form

u(r, s) ≈ M ∑ i=1 R i (r )S i (s). ( 29
)
The problem can therefore be solved in a separated representation as a sequence of problems in the r and s coordinate domains, by using the standard PGD procedure (see [START_REF] Chinesta | The Proper Generalized Decomposition for Advanced Numerical Simulations. a Primer[END_REF] for more details).

The main and major advantage of the proposed mapping is its capacity for preserving the separated form of Jacobians.

Numerical examples

This section addresses different case studies: 2D and 3D domains with non-separable interfaces, a parametric problem involving an inclusion and finally the problem of a thermal source moving along a one-dimensional domain. In all the numerical examples which follow, each domain is discretized with a uniform mesh.

2D domains with non separable interface

In this section the applied boundary conditions (in the sequel all units are assumed to be in the metric system) read: In what follows, we analyze two kinds of non-separable boundaries.

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ u(x, y = 0) = 0, u(x, y = H ) = 25, ∂u(x, y) ∂ y | x=0 = 0, ∂u(x, y) ∂ y | x=L = 0. (30)

Parabolic interphase

In this first numerical example we consider an extremely thin rectangular domain Ω = (0, 1) × (0, 10 -3 ), i.e., H = 10 -3 and L = 1, with the internal boundary h(x) defined by the parabolic curve

h(x) = H 2 ( 0.5 + 2x -x 2 ) . ( 31
)
This boundary defines two domains with different thermal conductivities: the upper domain has a conductivity K u = 10, whereas the bottom one has a conductivity K b = 1. The interface h(x) is depicted in Fig. 2. The PGD solution is achieved in 0.2 s on a standard laptop using 100 nodes in the r -mesh and 1000 nodes in the s-mesh. Note that this resolution is equivalent to the use of 10 5 finite element degrees of freedom.

The solution in the (r, s) domain is shown in Fig. 3, whereas the one in the actual (x, y) domain is depicted in Fig. 4. The PGD solution involves M = 6 terms, that allowed a residual reduction of 9 orders of magnitude, i.e., a reduction factor of 10 -9 .

When expressing the conductivity in a separated form according to Eq. ( 10) by invoking the singular value decomposition, K = 96 modes are needed for a reduction of six orders of magnitude of the associated eigenvalues. The solution using that K-mode separated representation of the thermal conductivity requires a non negligible effort, first for performing the SVD and then for constructing the separated representation of the temperature. In the last, more than 50 temperature modes (i.e., M > 50) were required to reach an acceptable accuracy, needing more than half an hour of calculation.

It is worth mentioning that, when using the approach based on the separation of the thermal conductivity, the usual separated representation constructor, see [START_REF] Chinesta | The Proper Generalized Decomposition for Advanced Numerical Simulations. a Primer[END_REF], that proceeds by computing rank-one updates, hardly converged, and each new mode produced only a very slight reduction of the residual.

Wavy boundary

In this second example the boundary h(x) is defined from

h(x) = H 4 sin (4xπ) + H 2 . ( 32
)
Meshes and model parameters are the same that the ones considered in the previous case study. Despite of poor separability of the thermal conductivity field, again by using the mappings discussed in Section 2, the solution contained only M = 6 modes. It was obtained in 0.36 s, with a similar residual reduction factor (10 -9 ). The solutions in the reference and actual domains, R and Ω , are depicted respectively in Figs. 5 and6. Again, the interface location is highlighted. When using a SVD-based separated representation of the conductivity field, the solver hardly converges, and every new mode reduced very slightly the residual. Consequently, the calculation required again more than half an hour.

Three-dimensional layered domain with an inclined flat interface

In this section the procedure described in Section 2 is extended to the 3D case considering the domain sketched in Fig. 7, with Ω = (0, L x ) × (0, H ) × (0, L z ).

⎧ ⎨ ⎩ x = r, y = s h(x, z), z = t, for s ∈ (1, 2) ⎧ ⎨ ⎩ x = r, y = (s -1) (H -h(x, z)) + h(x, z), z = t, (33)
where h(x, z) is given by The thermal problem is solved by prescribing temperatures at the top and bottom surfaces, u(x, 0, z) = 0 and u(x, H, z) = 25, while enforcing null heat fluxes on the lateral surfaces x = 0, x = L x , z = 0 and z = L z . The thermal conductivities of the subdomains located above and below the interface, K u and K b , are the same that the ones previously considered.

h(x, z) = (H/4) + (H/2) x L x . (34)
Fig. 8 depicts on the reference domain R the solution on the domain cross section t = L z /2, while Fig. 9 shows the solution at the corresponding cross section of the actual domain Ω . Fig. 10 depicts the 3D plot of the solution on the actual 3D domain Ω .

The meshes considered in this simulation consisted of 100 nodes along the r -and t-coordinates, and 1000 along the thickness s, to reach a resolution equivalent to 10 7 finite element degrees of freedom. The most remarkable fact was that the calculation was accomplished in 3 s in a standard laptop, and that M = 15 modes were enough for reducing the residual in 8 orders of magnitude, that is, a reduction by factor of 10 -8 .

When expressing the conductivity using both, an in-plane-out-of-plane separated form or a fully separated form, by invoking in both cases the SVD, the usual rank-one constructor [START_REF] Chinesta | The Proper Generalized Decomposition for Advanced Numerical Simulations. a Primer[END_REF] did not reach convergence after computing 100 modes (note that 15 modes were enough when using the procedure based on the mapping described in Section 2).

Parametric inclusion

In this example we analyze an issue of major relevance, related to a thermal problem in a one-dimensional domain x ∈ [0, L], with L = 1, assumed to be composed by a material of thermal conductivity K 1 = 5 that contains an inclusion of length l = L/20 and conductivity K 2 = 1 (IS units), with its central point located at position X ∈ [0.035L , 0.965L].

The parametric temperature field is sought. In other words, we look for the expression of the temperature field for any possible position of the inclusion X , i.e., u(x; X ). Within the PGD rationale, the inclusion location X is considered as an extra-coordinate, so that the temperature field becomes two dimensional, i.e. u(x, X ), while the complexity due to the increase of dimensionality is compensated by the use of the separated representation.

The parametric weak form of the heat equation now reads

∫ Ω ∂u * ∂ x K (x; X ) ∂u ∂ x d x = ∫ Ω u * Qd x, (35)
with Q the heat source term. We consider the mapping:

s ∈ [0, 1] { X = r, x = sh 1 (X ) = sh 1 (r ), s ∈ [1, 2] { X = r, x = (s -1)l + h 1 (X ) = (s -1)l + h 1 (r ), s ∈ [2, 3] { X = r, x = (s -2)h 2 (X ) + l + h 1 (X ) = (s -2)h 2 (r ) + l + h 1 (r ), (36)
where h 1 (X ) is the position of the inclusion left boundary, i.e. h 1 (X ) = Xl/2, l is the inclusion length and h 2 (X ) the distance between the right boundary of the inclusion and the domain right boundary, i.e. h 2 (X ) = L -Xl/2. The mapping (36) is sketched in Fig. 11. By using the expression of the Jacobian, the heat problem weak form (35) can be expressed in the reference (r, s) domain and discretized without major difficulties, thus leading the parametric solution in the separated form

u(r, s) ≈ M ∑ i=1 R i (r )S i (s).
(37)

First numerical case

In this first numerical example we consider Q = 50 with the thermal conductivity K (x; X ) expressed form:

K (x; X ) = { 5 if x ∈ [X -l/2, X + l/2], 1 if x ∈ [0, X -l/2] ∪ [X + l/2, L], (38)
and homogeneous boundary conditions, u(x = 0) = 0 and u(x = L) = 0. Meshes contained 1500 and 100 nodes along the sand r -domains, respectively. The parametric solution u(r, s), and consequently its counterpart u(x, X ), were obtained in less than 1s again by making use of a standard laptop, with a reduction factor of 30 with respect to the 100 1D calculations needed for computing the equivalent parametric temperature field, one for each of the 100 discrete positions of the inclusion. Of course, when increasing the problem dimensionality (2D or 3D) the computing time savings are much higher. The constructed separated representation involves M = 10 modes. Fig. 12 depicts the PGD solution of the problem while Fig. 13 illustrates the one related to the solution of the 100 1D problems. A maximum relative error of 0.7% was found. The maximum error localizes around the bottom left and top right corners, where the mapping exhibits the largest gradients.

It is worth noting, with respect to the discussion addressed in previous sections, that if we try, as discussed in [START_REF] Ch | Simulation avancée des problèemes thermiques rencontrés lors de la mise en forme des composites[END_REF], to proceed by separating the thermal conductivity K (x, X ) according to

K (x, X ) ≈ K ∑ i=1 F i (x)G i (X ), (39)
by invoking the SVD, the parametric solution u(x, X ) directly constructed in the domain Ω involves M = 97 modes, and requires 10 times more computing time.

Second numerical case

In order to better emphasizing the effect of the inclusion, we consider Q = 0, the thermal conductivity expressed now from

K (x; X ) = { 1 if x ∈ [X -l/2, X + l/2], 5 if x ∈ [0, X -l/2] ∪ [X + l/2, L], (40)
and the boundary conditions, u(x = 0) = 0 and u(x = L) = 25. The computational performance is almost the same as the previous numerical example. Fig. 14 depicts the PGD solution of the problem while Fig. 15 illustrates the one related to the solution of the 100 1D problems. Again a maximum relative error of 0.7% is found, in the regions where the mapping gradients are larger.

Thermal source moving along a one-dimensional domain

In this example, we consider L = 1 m and the thermal source location X defined in X ∈ Ω = [0.035L , 0.965L]. X thus depends on the time from X = x 0 + νt. In this example, the thermal source velocity ν corresponds to a movement from X = 0.035L to X = 0.965L in 1 s, leading to x 0 = 0.035 and ν = 0.93. The parametric weak form reads

∫ Ω×I u * ∂u ∂t + ∂u * ∂ x k ∂u ∂ x d xdt = ∫ Ω×I f (x, t)d xdt, (41)
where I represents the considered time interval. The source location and the geometrical construction is depicted in Fig. 16. The relation between time t and source location X writes

t = (X -x 0 ) ν . ( 42
)
From Fig. 16, the heat source can be easily defined from:

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ h 1 (X ) = X - h g 2 , h g (X ) = L 20 , h 2 (X ) = L - h g 2 -X. (43)
We define a new mapping:

For s ∈ [0, 1] { X = x 0 + νt, x = sh 1 (X ), For s ∈ [1, 2] { X = x 0 + νt, x = (s -1)h g + h 1 (X ), For s ∈ [2, 3] { X = x 0 + νt, x = (s -2)h 2 (X ) + h g + h 1 (X ), (44)
that allows transforming the problem weak form into the reference (physical) space (x, X ).

In the numerical solution, we assume the uniform source f = 1 defined between h 1 and h 2 as depicted in Fig. 16. The heat source term f (x, t) in the actual domain is shown in Fig. 17, associated to the one in the reference domain depicted in Fig. 18 for the apparent source intensity f app = f h g ν . The considered mesh consisted of 600 nodes along the s coordinate and 300 in the X domain. The PGD solution converged after 1.18 s and consisted of 15 modes, reducing the residual more than 3 orders of magnitude. The solution in the reference domain (X, s) is shown in Fig. 19, whereas the one in the actual domain (x, t) is depicted in Fig. 20.

The same problem was solved using the standard PGD procedure. First the source term was expressed in a space-time separated form by invoking the SVD. Because its poor separability, many modes were required, with the eigenvalues involved in the SVD decomposition decreasing very slowly. Then, the standard rank-one greedy algorithm computed the temperature solution. However, its poor separability required again more than 100 modes and more than 50 s calculation to find a solution that exhibits small oscillations.

Conclusions

In this paper we addressed the space separation in layered domains Ω where interfaces are not planar (or, even being planar, are not parallel with respect to the in-plane-coordinate). In these circumstances, former works stressed that space-separated representations lose their expected effectiveness. In fact, that conclusion was derived from the fact that these geometries involved too many terms in the material property-separated representations when invoking the SVD or its high order counterpart, the so-called HOSVD [START_REF] Kolda | Tensor decompositions and applications[END_REF]. In these circumstances, the separated representation constructor converges too slowly, with the subsequent impact on accuracy and computational efficiency.

In this work we proved that when the domain Ω is mapped into a fully separable one R standard separated constructors recover their efficiency, as the numerical results here described and discussed proved.

The technique here proposed opens numerous possibilities, such as the flow in thin ducts with rough surfaces, where fully 3D discretization fails, whereas the use of separated representations can lead to levels of resolution never attained before.

Fig. 1 .

 1 Fig. 1. Domain containing a nonplanar internal boundary.

Fig. 2 .

 2 Fig. 2. Interface in Ω defining the top and bottom domains Ω u and Ω b .

Fig. 3 .

 3 Fig. 3. Solution in the reference domain R = R u ∪ R b .

Fig. 4 .

 4 Fig. 4. Solution in the actual domain Ω = Ω u ∪ Ω b with the interface location highlighted.

Fig. 5 .

 5 Fig. 5. Solution in the reference domain R = R u ∪ R b .

Fig. 6 .

 6 Fig. 6. Solution in the actual domain Ω = Ω u ∪ Ω b with the interface location highlighted.

Fig. 7 .

 7 Fig. 7. 3D case study involving a non-planar interface in the plate domain 1 × 10 -3× 1 (units in meters).

Fig. 8 .

 8 Fig. 8. Temperature u(r, s, L z /2).

Fig. 9 .

 9 Fig. 9. Temperature u(x, y, L z /2).

Fig. 10 .

 10 Fig. 10. 3D solution representation in the plate domain of dimension 1 × 10 -3 × 1 m 3 .

Fig. 11 .

 11 Fig. 11. The geometrical transformation depicted in Eq. (36).

Fig. 12 .

 12 Fig. 12. First numerical example: Parametric solution u(x, X ) computed by using the separated representation constructor operating on mapped domain R.

Fig. 13 .

 13 Fig. 13. First numerical example: Solution for the different inclusion positions, u(x; X i ), i = 1, . . . , 100.

Fig. 14 .

 14 Fig. 14. Second numerical example: Parametric solution u(x, X ) computed by using the separated representation constructor operating on mapped domain R.

Fig. 15 .

 15 Fig. 15. Second numerical example: Solution for the different inclusion positions, u(x; X i ), i = 1, . . . , 100.

Fig. 16 .

 16 Fig. 16. Heat source in the (x, X ) physical space.

Fig. 17

 17 Fig. 17. The heat source in the actual domain (X, s).

.

  Fig. 17. The heat source in the actual domain (X, s).

Fig. 18 .

 18 Fig. 18. The heat source in the reference domain (s, t).

Fig. 19 .

 19 Fig. 19. Temperature in the reference domain (X, s).

Fig. 20 .

 20 Fig. 20. Temperature in the actual domain (x, t).

✩ This work has been supported by the Spanish Ministry of Economy and Competitiveness through Grant number DPI2015-72365-EXP and by the Regional Government of Aragon and the European Social Fund, research group T24 17R.


Translating Networks Assessing correspondence between network visualisation and analytics

Digital Humanities 2019, Utrecht, Netherlands Martin Grandjean (1) and Mathieu Jacomy (2) 1. University of Lausanne, Switzerland 2. Aalborg University, Denmark Network interpretation is a widespread practice in the digital humanities, and its exercise is surprisingly flexible. While there is now a wide variety of uses in different fields from social network analysis [START_REF] Ables | Using Archival Texts to Create Network Graphs of Musicians in Early Modern Venice, Digital Humanities[END_REF] to the study of document circulation metadata (Grandjean, 2016) or literature and linguistic data (Maryl and Elder, 2017), many projects highlight the difficulty of bringing graph theory and their discipline into dialogue. Fortunately, the development of accessible software (Bastian et al., 2009), followed by new interfaces (Rosa Pérez et al., 2018;Wieneke et al., 2016), sometimes with an educational dimension (Beaulieu, 2017;Xanthos et al., 2016), has been accompanied in recent years by a critical reflection on our practices (Weingart, 2011;Kaufman et al., 2017), particularly with regard to visualisation. Yet, it often focuses on technical aspects.

In this paper, we propose to shift this emphasis and address the question of the researcher's interpretative journey from visualisation to metrics resulting from the network structure. Often addressed in relation to graphical representation, when it is not used only as an illustration, the subjectivity of translation is all the more important when it comes to interpreting structural metrics. But these two things are closely related. To separate metrics from visualisation would be to forget that two historical examples of network representation, Euler (1736) and Moreno (1934), are not limited to a graphic reading (the term "network" itself would only appear in 1954 in Barnes' work). In the first case, the demonstration was based on a degree centrality measurement whereas in the second case the author made the difference between "stars" and "unchosen" individuals while qualifying the edges as inbound and outbound relationships. This is why this paper propose to examine the practice of visual reading and metrics-based analysis in a correspondence table that clarifies the subjectivity of the translation while presenting possible and generic interpretation scenarios.

Visual approach: making the global structure readable

The way we read networks has changed over time. Historically the question of network readability was asked in terms of aesthetic criteria. In the word of Jacob Moreno "the fewer the number of lines crossing, the better the sociogram". Even in the nineties, when giving birth to the modern layout algorithm, Früchterman and Reingold (1991) aimed at "minimizing edge crossings" and "reflecting inherent symmetry". However, these criteria do not seem so crucial to practices observed nowadays in digital humanities (and beyond). (Choinski and Rybicki, 2017). These usages reveal a different perspective in network visualisation where we expect the visual to translate underlying relational structures. It helps to give different names to these two different approaches. We call diagrammatic the perspective where the network is a diagram that we read by following paths. We do not want the edges to cross and we use aesthetic criteria to bring clarity. It was Moreno's perspective, and is still relevant to small networks and local exploration. Then we call topological the perspective where the network is a structure that we read by detecting patterns. We expect the visualisation to help us retrieve structural features like clustering or centralities. It is a common practice in digital humanities, more holistic and relevant to larger networks. Aside or in complement, classic data visualisation is also employed to visualise non-relational structures (node attributes, etc.).

In the topological perspective, a standard procedure is to assign nodes a position using a force-driven algorithm. This family of algorithms is known for displaying clusters that match a widely used measure of community detection, modularity clustering (Noack, 2009). Its translation remains however difficult to interpret locally, as we can never give a simple explanation for a node's position. Classic data visualisation also translates non-relational structures, by itself or combined with a relational perspective.

Objectifying the structure with metrics

Often opposed to visual interpretation, of which they would be a more objective and reliable representation, centrality measures have a history that goes back to more than half a century and shows that they are not immutable and require constant adaptation to usage Moreover, Freeman (1979) insists on the fact that the notion of "centrality" is the result of several intuitive conceptions. To remind that these metrics are based on "intuition" means to recognize that they have no meaning in themselves and that their interpretation must be rediscussed -and therefore translated -according to the context. This paper thus proposes to list and evaluate to which extent these metrics are applicable to humanities and social science data and can, if necessary, be "translated" into this language to complement visual analyses.

Global properties

Statistical analysis allows for comparing networks across multiple dimensions at once (Tank and Chen, 2017). For instance, comparing the number of nodes and edges of different graphs of the same type (Trilcke et al., 2016) can be a ranking tool that is directly translatable into natural language. In addition to that, studies suggest that density (the number of edges in relation to the number of nodes) is relevant to analyse character networks, especially when compared within a homogeneous collection (Evalyn and Gauch, 2018;Grandjean, 2015). This is also the case when measuring average path length (Trilcke et al., 2016).

Local properties

With regard to local measures, the degree (number of neighbouring nodes) is the simplest centrality, and the only one systematically used between the late 1950s and early 1970s, before the development of more diversified metrics (Freeman, 1979). Its simplicity allows for a transparent translation: in a literary network, for example, it counts the number of times one character speaks to another (Jannidis et al., 2016).

The notion of betweenness centrality disrupts the conception of what the "centre" of a network may consist of. Its ability to reveal structural elements bridging large, immediately visible clusters makes it popular in the social sciences since the emergence of Granovetter's concept of "weak ties" (Granovetter, 1973). Betweenness is very closely linked to the notion of circulation: it counts the shortest paths to detect intermediate "bridges" or "key passages" capable of opening or locking certain parts of the network (Tayler and Neugebauer, 2018). Depending on applications, these are therefore both positions of power and vulnerable places.

The closeness centrality allows to highlight the "geographical" middle of the graph. In networks of a certain density and when they are not divided into several distinct communities, the closeness is generally fairly evenly distributed and allows a good translation of the notions of "center" and "periphery".

For its part, the eigenvector centrality is quite complicated to translate since it works iteratively and is very much dependent on the structural context at short and medium range around a node. "Prestige" or "influence" centrality, named "power" centrality by its author (Bonacich, 1972), it qualifies a node's environment while operating in cascade: a well-connected node gives its neighbours a part of its authority capital, and so on. It is therefore particularly useful when trying to analyse the hierarchy of the nodes in a graph (Piper et al., 2017). The most well-known use of this measure is the backbone of the Google search engine: the PageRank algorithm (Brin and Page, 1998).

Towards mixed approaches

This contribution proposes a table of correspondence between the concepts of graph theory and the practice of visual network analysis in the social science and humanities. This effort must not be understood as a demarcationist attempt at telling the right method from the wrong. The "dictionary" is not exhaustive and only aims at helping to bridge two worlds that have more in common that what meets the eye. By focusing on translating methods, we want to stress that crossing points are real even though they do not come without issues, and thus require our methodological attention.

We also note that the analysis should not be limited to a catalogue of well-known methods (basic centralities, etc.) but that approaches combining several of those should be encouraged to obtain an optimal and innovative "translation". In this way, we could compare metrics (Escobar and Schauf, 2018) or combine them to establish rankings (Fischer et al., 2018;Grandjean, 2018: 328). Furthermore, the enrichment of the networks by means of categories that are not dependent on the structure, like the gender of individuals in a social network (Dunst and Hartel, 2017) or the discipline of projects in a scientometric analysis (Grandjean et al., 2017), allows to test translation and interpretation hypotheses by avoiding the blind approach of testing all possible graph metrics. Counting the nodes.

Network visual and topological patterns

COMPUTATIONAL CRITERIA

Counting the nodes.

"TRANSLATION"

The intuition of the size of a network is appropriate.

Global

Graph size (edges)

VISUAL PATTERN

There are few or many edges.

ISSUES

The total number is hard to estimate as soon as the graph is not a simple diagram anymore. The distribution of edges and their weight has an influence on the visual estimation. The difficulty to count edges visually is a known issue, and probably impossible to overcome. Following the edges between every couples of nodes. Can be used to describe the size, breadth or width of the network. But it can also be translated into an indicator of a small world situation.

Connectedness

VISUAL PATTERN

There must be only one component, not several groups of nodes disconnected from each other.

ISSUES

Depending on the layout, it is possible that "islands" hide in dense groups of nodes, but with a properly set forcedirected layout, the risk is marginal.

Possibly the easiest property to observe visually.

TOPOLOGICAL PATTERN

In a connected graph, the number of connected components must be one.

ISSUES

The notion of connectedness is more complicated for a directed graph than for an undirected graph.

USEFULNESS

The absence of edges between components is more remarkable if they contain many nodes. In many applied cases, connected graphs are artificially created by removing solitary nodes (frequent in messy extracted data).

SIMPLE DEFINITION

Is the graph a connected system where there is a path between every nodes?

HERMENEUTIC CRITERIA

Looking at empty areas (structural holes). Groups count as disconnected only if there are no edges between them.

COMPUTATIONAL CRITERIA

There must be a path between each pair of nodes.

"TRANSLATION"

The network is a continent, or, on the contrary, an archipelago.

Global

Clusters/ Communities USEFULNESS Useful for exploration. It is tempting to take the result of a cluster calculation as a given.

In some cases, it is interesting to compare these clusters with previously known groups (categories that do not depend on the structure obtained).

SIMPLE DEFINITION

What are the groups where nodes are more connected to each other?

HERMENEUTIC CRITERIA

Looking for groups of nodes, as visually dense and separated as possible.

COMPUTATIONAL CRITERIA Running a modularity clustering detection algorithm and looking at the obtained modularity.

"TRANSLATION"

The term cluster has become part of the common language, but we also like to talk about groups, communities or hubs. This notion of community is very directly related to the way in which the social sciences and humanities use the metaphor of the "network".

Global

Global or average clustering coefficient

VISUAL PATTERN
Clusters.

ISSUES

Triangles are easy to count visually in a small network, but the ratio between this result and the total number of potential triangles is impossible to calculate directly. Very difficult to count in dense graphs.

Graphs with clusters and/or visually dense tend to have a higher clustering coefficient. Gives an idea of the entanglement / intrication and the presence of a more localized density.

Local

Connectivity (degree)

VISUAL PATTERN

There are many links to the node.

ISSUES

In a dense image, it is not always obvious which edges converge to the node or just happen to pass through it visually. Sometimes, counting is also impractical.

TOPOLOGICAL PATTERN

Degree.

ISSUES

In a directed network, we also distinguish indegree (inbound link) and outdegree (outbound links). In that case, the degree is the sum of those and hence the number of links, not the number of neighbors.

USEFULNESS

The simplest form of centrality. In most cases, the degree shows information that is already known as part of the basic data and not dependent on the structure. This is why we often focus on the degree distribution.

SIMPLE DEFINITION

How well connected is a node / how many links it has / how many neighbors HERMENEUTIC CRITERIA

Counting the edges converging to that node.

COMPUTATIONAL CRITERIA Degree.

"TRANSLATION"

The basic intuition of the number of neighbors. In directed networks, interpretation varies greatly between in-and out-degrees: indegree is often the primary way to look at a hierarchy of nodes, because being "cited" is often a good proxy for authority/notoriety.

Local

Betweenness

VISUAL PATTERN

A bridge between clusters.

ISSUES

Many bridges look as expected, they connect over empty spaces. But sometimes, bridges are hidden in the complicated structure of the image. It is generally easier to see the bridging edges than the bridging nodes (however, most of the studies using betweenness centrality focus on bridging nodes).

TOPOLOGICAL PATTERN Betweenness centrality.

ISSUES

Note that the undirected version of the algorithm is often used even for directed graphs. It is the most used metric for detecting bridges, but it does not exactly meet the intuition.

USEFULNESS

The definition of bridge implemented by betweenness centrality meets both intuition of a bridge and of a center. Indeed both a bridge and the center of a star are things that, if removed, disconnect parts of the network. In that sense betweenness is also a "centrality".

SIMPLE DEFINITION

Being a bridge, connecting otherwise separated groups of nodes.

Removing that node would break many shortest paths.

HERMENEUTIC CRITERIA

Looking for an edge appearing through a (mostly) empty area between large groups of nodes.

COMPUTATIONAL CRITERIA

The score of betweenness centrality represents the number of shortest paths through a given node or edge.

"TRANSLATION"

The notion of bridge (but also link, gateway, broker or key passage) is very often used when applying network analysis to social or circulation issues. In some cases, it can represent a form of social capital because it describes a structural position of power (or vulnerability).

Closeness

VISUAL PATTERN

The geographical center of the graph.

ISSUES

The visual estimation of centrality is considered acceptable, but it remains an evaluation.

It is harder to find in very sparse graphs.

TOPOLOGICAL PATTERN Closeness centrality.

ISSUES

The undirected version is often used even for directed graphs.

USEFULNESS

There is no single implementation of centrality, but closeness centrality is the most aligned with the notion of a middle, a point that is in proximity to all the others. It looks at the structural distance to other nodes, and can be interpreted as such.

SIMPLE DEFINITION

Being in the middle of the network.

HERMENEUTIC CRITERIA Finding the barycenter (the center of the "land masses") of the graph.

COMPUTATIONAL CRITERIA

The score of closeness centrality is the average length of the geodesic distances to all the other nodes.

"TRANSLATION"

Excellent to describe the centre or the middle of a network, especially when the latter is described in topographical terms. Low values of this metric are very appropriate for the use of concepts which are the opposite of the center: the periphery, the margins, etc.

Local Prestige (eigenvector) The iterative nature of this notion makes it difficult to translate (and difficult to use in some contexts). It is confused with the notions of prestige, authority, influence and, sometimes, power and elites. This measure distinguishes nodes that are "well" connected (and not just "a lot"). Relates to the notion of assortativity. Presence of a path between the two nodes whose relationship is to be analyzed.

Local

Local clustering coefficient

ISSUES

Requires that we can follow the links in practice, which is possible only for small (undirected) networks and depends on the graphic settings. Finding a path can be difficult, and ensuring that this path is the shortest can be too difficult. However the visual distance is a loose approximation of the shortest path length. Following the series of edges from one node to another to find the shortest.

COMPUTATIONAL CRITERIA Geodesic distance, algorithms for shortest path detection.

"TRANSLATION"

Corresponds to the intuitive notion of distance in the graph structure. Note that this translation does not take into account the fact that nodes are not always aware of the steps between them and that the perceived distance is not always the shortest path. The term clique itself refers to social groups of individuals who knows each other. They can be translated as communities, neighborhoods, closed societies.

Local

Cliques

  Looking at recent papers in digital humanities, networks appear to have a wide range of usages. Their visualisations are either self-sufficient [fig. 1.a.] (Algee-Hewitt, 2018; Pino-Diaz and Fiormonte, 2018; Verhoeven et al., 2018; Marraccini, 2017), an optional help to understanding [fig 1.b.] (Colavizza et al., 2016) or strongly connected to the text. Some authors use them to highlight the position of a specific node [fig. 1.c.] (Moretti et al., 2016), to compare layouts [fig. 1.d.] (Sozinova, 2016) or the layout of the same graph in time [fig. 1.e.] (Wright, 2016). They may aim at visualising communities [fig. 1.f.] (Rybicki et al., 2018; Torres-Yepez and Zreik, 2018), mapping a general structure [fig. 1.g.] (Gao et al., 2017), tracking density patterns [fig. 1.h.] (Gao et al., 2018) or monitoring algorithms like modularity clustering [fig. 1.i.]

Fig. 1

 1 Fig. 1 Different contexts for network visualisation in DH2016, DH2017 and DH2018 abstracts.

Fig. 2

 2 Fig. 2 Various layouts do not follow a force-driven algorithm to make non-relational dimensions of the data explicit.

Fig. 3

 3 Fig. 3 Three levels of interpretation that can be articulated: visual analysis (examples top left), use of global metrics (examples bottom right) and use of local metrics (highlighted nodes).

  that complements density well and, like the latter, is useful for comparing similar networks with each other.

  the use of the graph as a research interface to test the relation of couples of nodes. Very close to the qualitative approach of the humanities, which are often focusing on a few individuals in the network.

  cliques, their size and distribution are metrics that are complementary to the clustering coefficients (local and global). They can be used as a more strict community detection algoritm.

  This table of correspondence between network analysis concepts and interpretations or "translations" is a work in progress. The authors propose this document to open a discussion on the most relevant translation scenarios and examples/references from the different disciplines applying these methods.

	NOTION	VISUAL ANALYSIS	COMPUTATIONAL ANALYSIS INTERPRETATIVE POTENTIAL
		VISUAL PATTERN	ISSUES
		There are few or	Though it is easy to
		many nodes.	have an estimation of
			the total number of
			nodes, visualization
			decisions (for
	Global	Graph size (nodes)	example, setting node sizes on a large scale) can make nodes with few connections difficult to see.

TOPOLOGICAL PATTERN Nodes count.

ISSUES

None. Note that in graph theory the count of nodes is referred to as "graph order" while the "graph size" refers to the count of edges. USEFULNESS Very basic information but useful when comparing networks. SIMPLE DEFINITION How many nodes are there in the graph? HERMENEUTIC CRITERIA

  centrality is notably adapted to directed networks, and can be related to the functioning of a search engine (the Page Rank principle was used in the first Google search engine) or a system where information flows.

		VISUAL PATTERN	ISSUES	TOPOLOGICAL PATTERN
		Proximity to well	This centrality is hard	Eigenvector centrality
		connected nodes	to see, though it	or Page Rank.
		(often inside a	correlates with other
		cluster).	forms of centrality
			that point to well
			connected nodes at
			the center of the
			graph.
	SIMPLE DEFINITION	HERMENEUTIC CRITERIA		COMPUTATIONAL
	Being connected to well connected nodes without necessarily having a large number of neighbors itself.	None, except if the size of the nodes is visually proportional to the degree centrality, which helps to identify nodes in the hubs' surroundings.		CRITERIA This is a score computed recursively. It flows from each node to its neighbors (following the direction of edges in a directed graph).

ISSUES

None. USEFULNESS This form of "TRANSLATION"