Efisio Manunta

Maja Becker

Matthew J Easterbrook

Vivian L Vignoles

Manunta Efisio

Clle

Economic Distress and Populism: Examining the Role of Identity Threat and Feelings of Social Exclusion

Keywords: populism, identity threat, identity motives, economic distress, social exclusion

HAL is a multi-disciplinary

emerged as key factors contributing to increased support for populist beliefs and politicians throughout the last decade. Specifically, the 2007 economic crisis had negative repercussions for people's actual and perceived economic status within society, fueling economic distress, whereas the migration crisis in Europe and the United States has provoked reactions against some progressivist societal changes (e.g., global values, openness towards differences, multiculturalism) that some perceive as a cultural threat. Thus, public opinion in western countries has questioned the legitimacy of several social groups related to political, economic (e.g., banks, multinational corporations), and/or cultural (e.g., journalists, academics) elites. Simultaneously, the populist thin ideology has spread in these countries.

Our current article focuses on the economic distress pattern. Employing a correlational design, we analyzed the relation between economic distress predictors and adhesion to the populist thin ideology, while exploring the potential role of motivated identity processes in explaining this pattern [START_REF] Vignoles | Identity motives[END_REF]. We propose a new model of the economic distress pattern in which the frustration of identity motives relative to economic-status-based identity [START_REF] Destin | Status-based identity: A conceptual approach integrating the social psychological study of socioeconomic status and identity[END_REF]-a form of identity threat-predicts adhesion to populist thin ideology. We further examined whether this identity threat in turn is predicted by indices of perceived economic deprivation. [START_REF] Mudde | Populism: An ideational approach[END_REF] offered an exhaustive definition of populism as a thin ideology composed of two components: (1) that society is divided into a "good people" ingroup and a "corrupted élite" outgroup, and (2) that politics should be the direct expression of a people's general will without any institutional mediation. Mudde's theorization of populism inspired research considering populism as a set of specific measurable beliefs: the ideational approach. This approach has highlighted several individual-level predictors of populist endorsement, which refer to both the economic and the cultural threats often emphasized in theoretical literature on the link between populism and economic and cultural crises [START_REF] Inglehart | Trump, Brexit, and the rise of populism: Economic have-nots and cultural backlash (HKS Working Paper No RWP16-026[END_REF].

THE IDEATIONAL APPROACH TO THE STUDY OF POPULISM

THE LINK BETWEEN ECONOMIC DISTRESS AND POPULISM

A commonly identified explanation for recent increases in populism in Europe and western democracies is economic distress. The underlying hypothesis is that the current wave of populism was originally triggered by the 2007-08 economic crisis and the subsequent economic structural changes that disrupted western societies and disintegrated their middle classes [START_REF] Albertini | The relation between social class and economic inequality: A strengthening or weakening nexus? Evidence from the last three decades of inequality in Italy[END_REF][START_REF] Schettino | Income polarization in the USA: What happened to the middle class in the last few decades?[END_REF]. Evidence in social and political sciences has supported the validity of this hypothesis at a macro level [START_REF] Corbet | Populism and extremism: The immediate political challenges to Europeanism[END_REF] and found that macro indices of economic disparity are related to support for populism at both regional and national levels [START_REF] Oxendine | The political psychology of inequality and why it matters for populism[END_REF].

Highlights

• An integrative model of economic distress and identity threat predicts support for populist thin ideology.

• Findings provide original evidence of the association between economic status-based identity threat-the frustration of identity motives-and support for populist thin ideology.

• The frustration of belonging-feelings of social exclusion-was the one identity motive that robustly predicted endorsement of the populist thin ideology.

• Three different levels of relative deprivation (RD)-intergroup, interindividual, and temporal-were directly or indirectly involved in the economic distress pattern: Intergroup RD was the strongest predictor of populism.

Political and social psychology have investigated the relation between economic distress and populist endorsement at the individual level. Subjective economic and/or socioeconomic status indicators [START_REF] Adler | Relationship of subjective and objective social status with psychological and physiological functioning: Preliminary data in healthy, White women[END_REF] and experiences of relative deprivation (RD) have been found to predict populism [START_REF] Marchlewska | Populism as identity politics: Perceived in-group disadvantage, collective narcissism, and support for populism[END_REF][START_REF] Urbanska | Swaying to the extreme: Group relative deprivation predicts voting for an extreme right party in the French presidential election[END_REF]. Although subjective social status and RD are both indices of economic distress, the two constructs differ substantially from one another. The former is the economic and social position that someone perceives that they are in, whereas the latter is the extent to which they feel economically deprived in relation to a specific comparator. The RD construct represents an individual's perception of their economic situation in relation to one of several possible comparators, including intergroup (other groups in society), interindividual (other individuals), and/or temporal (one's past situation) comparison targets [START_REF] Smith | Relative deprivation: A theoretical and metaanalytic review[END_REF]. Furthermore, RD includes both a cognitive and an affective dimension. The cognitive dimension taps into the extent to which individuals perceive that their own position is disadvantageous compared to the comparator. The affective dimension taps into the negative emotional reaction to the perception of being relatively deprived in relation to the comparator, specifically feelings of injustice and anger. Following Smith and coauthors' meta-analysis on RD, a good operationalization should include both cognitive and emotional dimensions.

Thus, in the present study we considered subjective social status as a potential predictor of RD. The perception of social status could be expected to directly influence perception and feelings of RD, such that low-status individuals would feel more deprived than high-status individuals. As we discuss next, we expected that economic status-based identity threat would partially or wholly mediate the relationship of these economic threat variables with endorsement of populism.

ECONOMIC STATUS-BASED IDENTITY THREAT

Although the effect of identity threat on individuals' adhesion to the populist thin ideology has received some theoretical attention [START_REF] Hogg | Self-uncertainty and populism: Why we endorse populist ideologies, identify with populist groups, and support populist leaders[END_REF], there are as yet few empirical studies supporting this assumption. The present studies aim to fill this empirical gap. We analyzed identity threat-in the form of frustration of identity motives-in relation to the perception of one's identity regarding their economic position, informed by motivated identity construction theory (MICT; [START_REF] Vignoles | Identity motives[END_REF].

MICT [START_REF] Vignoles | Identity motives[END_REF] suggests that, beyond social identity theory's emphasis on self-esteem and distinctiveness [START_REF] Tajfel | An integrative theory of intergroup conflict[END_REF], at least four other identity motives are involved in the identityconstruction process: belonging, continuity, meaning, and efficacy. The motive for self-esteem is the need to perceive oneself positively; belonging is the need to perceive oneself as accepted by important others and one's social groups; distinctiveness is the need for a sense of identity that is clearly distinguishable from others; continuity is the need to perceive that there is a continuity between the past, present, and future projections of the self; meaning is the need to perceive that one's existence matters in the grand scheme of things; and efficacy is the need to perceive oneself as able to act and impact one's environment. The frustration of these psychological motives induces identity threat and is associated with negative emotions, depressive states, and other forms of mental health conditions. Furthermore, identity motives have been shown to drive self-regulation: [START_REF] Vignoles | Beyond self-esteem: Influence of multiple motives on identity construction[END_REF] found that individuals tend to perceive aspects of the self that satisfy identity motives as more important or self-defining, whereas they perceive identity aspects that frustrate identity motives as more peripheral to self-definition.

MICT argues that identity motives are relevant to both personal and collective identities and that any aspect of one's life that satisfies the identity motives can become a salient identity element within one's self-definition [START_REF] Vignoles | Identity: Personal and social[END_REF]. Accordingly, in this study we made no marked distinction between collective (social) and personal (individual) identity. The identity element that we considered in our two studies was that related to their economic position in society. The status-based identity has been conceived as a specific identity content, which is the outcome of an integrative narrative, social and future-oriented identity-construction process [START_REF] Destin | Status-based identity: A conceptual approach integrating the social psychological study of socioeconomic status and identity[END_REF].

Thus, this identity combines collective and individual levels: It might be perceived both in relation to other group members sharing this peculiar condition and as an element reflecting their individual position. Both are likely to incorporate changes in the past as well as perceived chances of improvement or decline in the future.

Here, the economic status-based identity element was made salient in order to determine the extent to which it frustrates (or satisfies) the identity motives. We investigated the frustration of identity motives in general (Study 1)-and, in particular, the belonging motive (Study 2)-relative to economic status-related identity as a predictor of the populist thin ideology and as a mediator between the economic distress pattern and populism.

THE PRESENT STUDIES: INTEGRATING IDENTITY THREAT TO THE ECONOMIC DISTRESS PATTERN OF POPULISM

Recently, scholars have proposed that economic distress may be an antecedent to populism [START_REF] Inglehart | Trump, Brexit, and the rise of populism: Economic have-nots and cultural backlash (HKS Working Paper No RWP16-026[END_REF][START_REF] Oxendine | The political psychology of inequality and why it matters for populism[END_REF][START_REF] Rhodes-Purdy | Populist psychology: Economics, culture, and emotions[END_REF]. The underlying assumption is that economic crises threaten the basic motivational pattern that is required to make individuals perceive that democracies are the most appropriate form of government. According to [START_REF] Bar-Tal | Socio-psychological analysis of the deterioration of democracy and the rise of authoritarianism: The role of needs, values, and context[END_REF], democratic systems endure as long as they are perceived as able to satisfy citizens' epistemic and security needs. This is more generally related to the fact that economic and social inequality threats account for a variety of motivated collective and individual reactions to satisfy need for control and need for self-esteem [START_REF] Fritsche | The consequences of economic threat for motivated social cognition and action[END_REF].

In this paper, we aimed to integrate economic distress factors with motivated identity-construction processes [START_REF] Vignoles | Identity motives[END_REF] to pursue a better understanding of populism. We therefore integrated subjective socioeconomic status, RD, and identity threat into a single model predicting support for populism and populist voting.

We conducted two studies on the French population aiming to analyze the role of frustration (versus satisfaction) of identity motives as a predictor of endorsing the populist thin ideology. In Study 1, we focused on the role of identity threat including the frustration of all six identity motives proposed in MICT. In Study 2, we built on the results of Study 1 by focusing more closely on the role of social exclusion (frustration of the belonging identity motive) and its effects on populist beliefs.

These objectives were expressed in six hypotheses (see Figure 1). H1: Subjective social status was expected to be a negative predictor of three levels of RD (intergroup, interindividual, and temporal) and of economic-based identity threat (identity motives frustration). H2: Intergroup, interindividual, and temporal RD would be positive predictors of the adhesion to the populist thin ideology. H3: The three levels of RD were expected to be positive predictors of the frustration (versus satisfaction) of identity motives. H4: Economic-based identity threat would be a positive predictor of the populist thin ideology. H5: Positive indirect effects of the three forms of RD on populism through identity threat (mediator) were expected. H6: Populist thin ideology endorsement was expected to be a positive predictor of populist vote intentions.

In this model, subjective social status is an antecedent of RD and economic-based identity threat but is not expected to have a direct effect on populism. This is because subjective social status concerns one's perceived position in society and is not an index of threat on its own [START_REF] Adler | Relationship of subjective and objective social status with psychological and physiological functioning: Preliminary data in healthy, White women[END_REF]. Rather, we expected populism to be directly predicted by threat indicators (i.e., RD and status-related identity threat). RD was distinguished at the intergroup, intraindividual, and temporal levels, three of the possible object comparisons proposed by [START_REF] Smith | Relative deprivation: A theoretical and metaanalytic review[END_REF]. These levels were expected to be correlated but not redundant.

Populist vote intention was modeled as the final outcome predicted by the populist thin ideology. French politics at the time of data collection was an appropriate context for studying populism. Indeed, both right-and left-leaning populism has showed strong vitality throughout the last decade in France: For instance, Jean-Luc Mélenchon (La France Insoumise) is considered a left-wing populist leader, and Marine Le Pen (Le Rassemblement National) is considered a right-wing populist leader [START_REF] Ivaldi | Populism in France[END_REF]. Both leaders exceeded 15% of votes at the first round of the presidential election in 2017 and were considered compelling candidates in the presidential election in 2022.

The focus of the present research was on the economic distress pattern within the study of populism, examining the potential role of economic status-related identity threat in explaining this pattern. Although we were not looking to test a comprehensive predictive model of populism, we controlled for expected effects of four additional variables in order to strengthen results relative to our specific focus. Based on the cultural threat pattern, we expected perception of anomie (C1) and national collective narcissism (C2) to be significant positive predictors of populism [START_REF] Elçi | Politics of nostalgia and populism: Evidence from Turkey[END_REF][START_REF] Hartwich | Exposure to neoliberalism increases resentment of the elite via feelings of anomie and negative psychological reactions: Neoliberalism and anti-elitism[END_REF][START_REF] Lammers | Make America gracious again: Collective nostalgia can increase and decrease support for right-wing populist rhetoric[END_REF][START_REF] Marchlewska | Populism as identity politics: Perceived in-group disadvantage, collective narcissism, and support for populism[END_REF][START_REF] Oliver | Rise of the Trumpenvolk: Populism in the 2016 Election[END_REF]. National collective narcissism is the belief that one's national ingroup is somehow better than other nations (Golec de [START_REF] Golec De Zavala | Collective narcissism and its social consequences[END_REF]. Perceived anomie is the perception that society and its traditional points of reference-such as morality, rules, predictabilityare eroding, so that it is difficult to feel properly oriented into society [START_REF] Hartwich | Exposure to neoliberalism increases resentment of the elite via feelings of anomie and negative psychological reactions: Neoliberalism and anti-elitism[END_REF][START_REF] Sprong | Our country needs a strong leader right now": Economic inequality enhances the wish for a strong leader[END_REF][START_REF] Teymoori | Revisiting the measurement of anomie[END_REF].1 Lastly, right (versus left) political orientation on economic (C3) and societal/moral issues (C4) were expected to show a negative and a positive effect on the populist thin ideology, respectively (Manunta & Becker, manuscript under preparation).

STUDY 1 Methods

Sample

Study 1 was conducted on a sample (n = 458) of the French adult population aged from 18 to 81 years (M = 40.22, SD = 13.42). We excluded from our analyses respondents who declared not being French citizens and did not fully complete the survey. The sample was composed of 63.97% women, 34.72% men, 1.31% nonbinary or other genders. Data were collected between the March 17 and May 22, 2020. During this period, the entire French territory was under strict lockdown due to the Covid-19 pandemic: Schools and nonessential services were closed, nonessential travels and gatherings banned, and remote working was mandatory. This period also corresponded to the end of the two-year-long Yellow Vest crisis.

Instruments and Procedure

Participants were invited to respond to an online survey and were mostly recruited by announcements on social networks. The survey was composed by a set of items referring to socialpsychological and/or political measures (intergroup, interindividual, and temporal RD, identity motives threat, collective narcissism, anomie, populism thin ideology, political position on social and economic issues, populist vote), subjective social status, and sociodemographic measures (age, education, and gender). The order of the measures was randomized but sociodemographic variables and subjective social status always appeared after political-and social-psychological ones.

Identity Threat (Identity Motives Frustration Versus Satisfaction)

Each identity motive was measured by a balanced 4-item scale (2 reversed items per motive), self-esteem, efficacy, belonging, distinctiveness, continuity, and meaning. This scale was specifically created for this study, inspired by other ones used in previous studies into identity motives [START_REF] Easterbrook | Different groups, different motives: Identity motives underlying changes in identification with novel groups[END_REF][START_REF] Thomas | Social identification in sports teams: The role of personal, social, and collective identity motives[END_REF]. In Study 1, we were interested in the general identity threat, so we treated the frustration of all identity motives indistinctly, as part of a single identity-threat variable. The internal coherence was satisfactory (Ω h = .71) and was calculated following recommendations for variables that present an underlying multidimensional structure [START_REF] Viladrich | Un viaje alrededor de alfa y omega para estimar la fiabilidad de consistencia interna[END_REF]. Response scales were formulated in a 7-point Likert format requiring respondents to express agreement versus disagreement towards statements such as "My economic position in society makes me feel excluded" (one of the items for frustration of belonging motive). All items highlighted the status-based identity [START_REF] Destin | Status-based identity: A conceptual approach integrating the social psychological study of socioeconomic status and identity[END_REF] focusing on the economic position occupied in society, in order to explicitly measure how the perception of this particular identity content frustrates (versus fulfils) identity motives. Scores were calculated such that the higher the score, the higher the frustration/threat (versus satisfaction). For analyses using observed scores, we computed the sum of all 24 items. For analyses using identity threat as a latent variable, we treated the sums for each of the six motives as observed indicators. In a set of exploratory analyses, we further used this scale to consider the frustration of each identity motive separately (following [START_REF] Vignoles | Beyond self-esteem: Influence of multiple motives on identity construction[END_REF][START_REF] Vignoles | Identity motives[END_REF]. Populism and Motivated Identity Construction

Relative Deprivation

RD was divided into three dimensions that were treated distinctly in our analyses: intergroup (Ω = .88), interindividual (Ω = .81), and temporal RD (Ω = .92). The intergroup level consists in RD perceived for one's ingroup in relation to outgroups; the interindividual level consists in the individual perception of RD compared to other members of the same economic ingroup; finally, the temporal level refers to the perception that one's economic resources were higher in the past than the present. Each dimension was composed of 4 items (2 of which reversed) inspired by [START_REF] Smith | Relative deprivation: A theoretical and metaanalytic review[END_REF] metaanalysis that offered an in-depth conceptualization of RD. As suggested by Smith and colleagues, every dimension was composed of items measuring both the perceived extent of the economic difference with the object of comparison (intergroup, interindividual, or temporal) and emotions related to the perception of this difference. Example items for each dimension: "In general, to what extent do you think that your social class has fewer economic resources than the other social classes in society?" (intergroup RD); "To what extent do you feel that the difference in economic status between you personally and other people in your social class is unfair?" (interindividual RD); and "How satisfied are you with the difference between your personal economic condition in the past and the present?" (temporal RD). Participants responded using 7-point scales, and points were attributed in such a way that higher scores corresponded to higher RD levels. Both observed scores (sum of the items) and latent variables (with items as indicators) were employed for each depending on the kind of statistical analysis.

Subjective Social Status

Subjective social status was measured by the ladder measure of Subjective Social Status [START_REF] Adler | Relationship of subjective and objective social status with psychological and physiological functioning: Preliminary data in healthy, White women[END_REF]. This is a measure of the perceived socioeconomic position in society. Responses were coded so that the higher the score, the higher the perceived level of social status (1 = the bottom of society; 7 = the top of society).

National Collective Narcissism (Towards French National Group)

We used the 5-item version of the collective narcissism scale (Golec de [START_REF] Golec De Zavala | Collective narcissism and its social consequences[END_REF]; Ω = .78) translated in French [START_REF] Bertin | Stand out of my sunlight: The mediating role of climate change conspiracy beliefs in the relationship between national collective narcissism and acceptance of climate science[END_REF], and with items referring to the French national group (e.g., "If French people had a major say in the world, the world would be a much better place"). Participants expressed their level of dis/agreement using 7-point Likert scales. Scores were calculated so that higher scores corresponded to higher collective narcissism levels. Both observed scores and latent variables were employed in our analyses.

Anomie

We used a 6-item (Ω = .86) scale with three items taken from Elchardus and Spruyt (2016)'s scale (e.g., "I no longer understand what is happening in the world today"). The remaining three items were reversed items created by us specifically for this study. Participants expressed their level of dis/agreement on 7-point Likert scales. Scores were calculated so that higher scores corresponded to higher levels of anomie. Both observed scores and latent variables were employed in our analyses.

Right (versus left) Political Orientation

We used two self-placement measures of political orientation [START_REF] Lesschaeve | The predictive power of the left-right self-placement scale for the policy positions of voters and parties[END_REF]. Right (versus left) economic concerned right-left orientation regarding economic politics issues ("Please indicate your political beliefs, between left and right, regarding economic issues [such as social benefits, government spending, tax cuts]"). Right (versus left) social concerned right-left orientation regarding societal/moral issues. Responses were given on 7-point response scales (1 = left and 7 = right).

Sociodemographics

Data on age and education levels were collected using multiple response items. Education was measured by the highest education degree obtained. Following the French education system, seven response options were provided (1 = no education degree; 7 = master or above). Gender was measured as a binary variable (1 = male, 2 = female; the response "other/I don't recognise myself on the binary spectrum" was also proposed in the survey but treated as a missing value in the analyses given a low frequency).

Populist Thin Ideology

The populist thin ideology was measured by the POPulist Thin Ideology Scale (POP-ThIS, Manunta & Becker, manuscript under preparation). This is composed of 10 Likert scale items (Ω = .88; 7-point response scales), half of which are reversed. Two example items are "Banks, politicians and all the establishments belong to the same corrupt caste"; and "The Elite is not more corrupt than the People" (reversed). Scores were coded such that higher scores corresponded to higher levels of populism. This was used as both observed and latent variable depending on the analyses.

Populist Vote Intention

Populist vote intention was measured by asking respondents to think about their own upcoming potential vote. Participants were asked to indicate for which political party they would vote. These voting intentions were then transformed into a dichotomous variable (1 = populist; 0 = nonpopulist) drawing on a categorization of French electoral lists in the last European elections in 2019 provided by three independent raters. The categorization between populist and not populist parties was independent from the right-left axis. Raters were asked to categorize between populist and not populist parties no matter their position on the right-left axis by taking into account the kind of political discourses led by the members of each party. As a result, both right and left parties were categorized as populist (e.g., for the left: La France Insoumise; for the right: Le Rassemblement National), as well as both right and left parties were categorized as not populist (e.g., for the left: Génération.s, for the right: Les Republicains). Raters were French master's students in political sciences. Using this measure, the conditions in which participants expressed their vote intention were very similar to the actual conditions in the voting booth (secrecy, confidentiality, etc.). It might therefore be considered a proxy for the corresponding behavioral variable.

Analytical Steps

This study drew on a cross-sectional design (correlation matrix: Table S1 in the online supporting information). Beyond the already mentioned descriptive statistics and internal coherence estimations, we conducted linear regressions and structural equation models (SEM). First, multiple linear regression with observed measures was used to test Hypothesis 3 and Hypothesis 4 controlling for anomie, collective narcissism, right (versus left) economic and social, and sociodemographics. Second, a SEM with the expected regression paths was employed to test all the hypotheses simultaneously, as shown in Figure 1. All the hypothesized direct effects on populism were expected to be significant in both the regressions and the SEM to be corroborated. Variables for which the direct effect on populist thin ideology was expected but not found were excluded from SEMs, unless they were expected to be involved in other paths. Additional regressions and structural equation models were performed for exploratory purposes.

Since all hypotheses were expressed with specific relation (positive or negative), respective t-tests for Beta indices presented below correspond to the one-tailed versions. In contrast, t-tests for Beta indices relative to paths for which no specific hypotheses were formulated were performed in their two-sided versions. Hypotheses were corroborated when p < .05. Finally, all the analyses were conducted with R v 4.0.2 (R Core Team, 2020) and its packages lavaan 0.6.-7 [START_REF] Rosseel | lavaan: An R package for structural equation modeling[END_REF] and psych 2.0.7 [START_REF] Revelle | psych: Procedures for psychological, psychometric, and personality research[END_REF].

Results

Direct Effects on Populism

First, a multiple linear regression (Table S2 in the online supporting information) predicting the populist thin ideology was run to test the effects of identity threat (β = .14, p = .002, 95% CI [.06, .22]), intergroup RD (β = .16, p < .001, 95% CI [.08, .24]), interindividual RD (β = .09, p = .010, 95% CI [.03, .16]), and temporal RD (β = .16, p < .001, 95% CI [.09, .24]). Controls were also included: national collective narcissism (β = .09, p = .019, 95% CI [.02, .16]), anomie (β = .01, p = .443, 95% CI [-.06, .07]), right-left position for economic politics (β = -.39, p < .001, 95% CI [-.46, -.32]), right-left position for social/moral politics (β = .09, p = .019, 95% CI [.02, .16]), age (β = -.04, p = .362, 95% CI [-.12, .04]), education (β = .00, p = .978, 95% CI [-.08, .09]), gender (β = -.08, p = .033, 95% CI [-.16, -.01]), and subjective social status (β = -.02, p = .613, 95% CI [-.12, .07]). The regression accounted for approximately 37% of variance in populist thin ideology.

Testing Our Model

A structural equation model was performed as described in Figure 2. Anomie was excluded from this model because the expected direct effect was not found in the previous regression analysis. The model showed satisfactory fit (Χ 2 [df = 604, N = 380] = 1,596.91, p < .001, robust CFI = .904, robust RMSEA = .075[.072, .077]). Structural paths from the model are shown in Figure 2: Expected direct effects on populism were similar to the above-described regression analysis. The direct effects of subjective social status on the other dimensions of the "economic distress" pattern were confirmed for intergroup RD (β = -.54, p < .001, 95% CI [-.60, -.47]), temporal RD (β = -.42, p < .001, 95% CI [-.49, -.35]), and identity threat (β = -.33, p < .001, 95% CI [-.42, -.24]), but not for interindividual RD (β = -.02, p = .381, 95% CI [-.11, .07]). Indirect effects of intergroup (β = .02, p = .039, 95% CI [.002, .050]), interindividual (β = .05, p = .005, 95% CI [.02, .08]), and temporal RD (β = .02, p = .034, 95% CI [.002, .042]) on populist thin ideology were significant. The direct effect of the populist thin ideology on populist vote intentions was significant (β = .72, p < .001, 95% CI [.65, .79]).

Exploratory Analyses

We conducted an additional set of analyses (Table S4 in the online supporting information and Figure 3) to explore the roles of the six separate identity motives as predictors of populism. First, a two-block linear regression analysis was conducted to explore the role of each identity motive as a predictor of the populist thin ideology distinctly. This first regression block, including frustration of the six identity motives as parallel predictors, showed that frustration of belonging was the only identity motive that significantly predicted populist thin ideology (β = .26, p < .001, 95% CI [.15, .37]). In the second regression block, controlling for anomie, collective narcissism, right-left axes, and sociodemographics, the effect of frustration of belonging remained significant (β = .18, p = .001, 95% CI [.08, .28]) and was the identity motive with the strongest effect; an additional effect of continuity became significant in this model (β = .10, p = .026, 95% CI [.01, .19]).

Consequently, we replicated the same structural equation model run previously to test the hypotheses (Figure 2) but replacing (latent) identity threat with a latent measure based only on the 4 items regarding the frustration of the belonging motive (social exclusion). This SEM (Figure 3) yielded very similar results to the previous one. First, it showed satisfactory fit (Χ 2 [df = 538, N = 380] = 1,423.57, p < .001, robust CFI = .902, robust RMSEA = .075 [.072, .078]). In this model, the frustration of belonging motive had a significant effect on the populist thin ideology (β = .18, p = .003, 95% CI [.07, .30]) and was a significant partial mediator between the three forms of RDintergroup (indirect effect: β = .03, p = .024, 95% CI [.01, .06]), interindividual (indirect effect: β = .04, p = .014, 95% CI [.01, .06]) and temporal RD (indirect effect: β = .05, p = .012, 95% CI [.01, .09])-and endorsement of the populist thin ideology.

STUDY 2

Study 2 was a replication of Study 1, carried out to confirm the exploratory findings following which the frustration of belonging motive was a direct predictor of the populist thin ideology and a mediator in the economic distress pattern. Results from Study 1 were consistent with previous findings that social exclusion exacerbates ingroup/outgroup distinctions and similarities between ingroup members [START_REF] Bernstein | The impact of race and inclusionary status on memory for ingroup and outgroup faces[END_REF][START_REF] Sacco | The world in black and white: Ostracism enhances the categorical perception of social information[END_REF]. Since the populist thin ideology is characterized by the perception of an intergroup bias opposing the ingroup people versus the outgroup élite [START_REF] Obradović | Understanding the psychological appeal of populism[END_REF], the purpose of this study was to test the role of feelings of social exclusion (i.e., threat to or frustration of the belonging identity motive) as a predictor of adhesion to the populist thin ideology and as underlying key-factor of the economic distress pattern. This purpose was operationalized in the same set of hypotheses as those of Study 1, only replacing the global identity threat with the more specific threat to the belonging identity motive (Figure 4, H4). Also, the effect of anomie (C1, Figure 1), which failed to reach significance in Study 1, was not expected in Study 2. Hypotheses were preregistered 2 before data collection was launched (preregistration).

Based on theoretical arguments combined with the findings in Study 1, we designed Study 2 to conduct a confirmatory analysis regarding the role of frustration of belonging in the economic distress pattern. Nevertheless, in addition to the frustration of belonging, the frustration of continuity was also found to be a significant predictor of the populist thin ideology in Study 1, but this effect was unstable-that is, it was not significant in the first block of the regression in Table S4 in the online supporting information. Thus, we also conducted, and present below, further exploratory analyses of continuity as a predictor of populism, in view of future research.

Methods

Sample

Study 2 was conducted on a sample (n = 1,050) of the French adult population aged from 18 to 90 years (M = 48.86, SD = 15.51). The only inclusion criterion was to be a French citizen. Respondents who were not French or did not fully complete the survey were excluded from our analyses. The sample was composed of 63.71% females, 33.91% males, and 2.38% nonbinary or other genders. Data were collected between January 25 and February 11, 2021. During this period, restrictions related to the COVID-19 pandemic in France were less strict compared to those during Study 1. There was no general lockdown, schools were open, remote working was recommended but not mandatory, and indoor private visits were not banned even though meeting outdoor was encouraged. Some restrictions were active: Indoor services were not allowed for bars and restaurants and a night curfew was implemented. 2 Numbering of hypotheses differed between paper and preregistration. The effects of subjective social status in the preregistration were expected to be positive because we anticipated scoring this variable in the opposite direction (the higher the score the lower the economic position), but this was reversed for a better understanding after reviewers' suggestion, thus reversing also the direction of the stated relation in the hypothesis. Due to an error, the paths Hypothesis 1b, Hypothesis 1d, and Hypothesis 3b (Figure 4) were not preregistered. Nevertheless, we decided to include them because there were no theoretical arguments for their exclusion. These paths were consistent with findings in Study 1 and the theoretical sense of the preregistered model. Populism and Motivated Identity Construction

Instruments and Procedure

Analytical Steps

This study drew on a cross-sectional design (correlation matrix: Table S5 in the online supporting information). The analysis conducted in this study was the exact replication of the structural equation model performed in the exploratory analyses of Study 1, only using the more focused measure of identity threat in the form of threat to belonging (social exclusion). This SEM with the expected regression paths was employed to test all the hypotheses simultaneously. T-tests' p-values for Beta indices presented for this structural equation model correspond to one-tailed versions.

A multiple linear regression on the populist thin ideology, including all identity motives distinctly and other control factors, was run to explore further evidence about the role of continuity to predict populist thin ideology.

Results

Testing the Hypotheses by the Structural Equation Model

The SEM (Figure 5) showed satisfactory fit (Χ 2 [df = 538, N = 860] = 2,004.19, p < .001, robust CFI = .936, robust RMSEA = .06. [.06, .06]). All results were similar to the corresponding model in Study 1. The direct effects of subjective social status on the dimensions of the "economic-identity threat" pattern were confirmed for intergroup RD (β = -.50, p < .001, 95% CI [-.55, -.45]), interindividual RD (β = -.18, p < .001, 95% CI [-.24, -.13]), temporal RD (β = -.37, p < .001, 95% CI [-.42, -.32]), and frustration of belonging (β = -.26, p < .001, 95% CI [-.32, -.21]). Indirect effects of intergroup (β = .02, p = .010, 95% CI [.01, .04]), interindividual (β = .03, p = .012, 95% CI [.01, .05]), and temporal RD (β = .02, p = .033, 95% CI [.002, .029]) on populism through the frustration of belonging were significant.

Expected direct effects on the populist thin ideology were confirmed for: frustration of belonging/social exclusion (β = .12, p = .006, 95% CI [.04, .19]), intergroup RD (β = .42, p < .001, 95% CI [.33, .50]), right-left position on economic issues (β = -.39, p < .001, 95% CI [-.45, -.34]), and collective narcissism (β = .30, p < .001, 95% CI [.24, .36]). Moreover, the direct effect of the populistthin-ideology scale on the populist voting intentions was significant (β = .69, p < .001, 95% CI [.64, .74]). Interindividual, temporal RD, and right (versus left) positions on the social/moral political issues did not show significant direct effects on populism.

Exploring the Role of Continuity and Other Identity Motives

A multiple linear regression of the populist thin ideology was run. In the first block, only the frustrations of each identity motive were included as factors (Table S7 in the online supporting information). In the second block, controls were added. Frustration of continuity showed a significant effect neither in the first (β = -.01, p = .643, 95% CI [-.08, .05]) nor in the second block (β = .00, p = .936, 95% CI [-.06, .06]). Only the frustration of belonging showed a significant positive effect in both the first (β = .30, p < .001, 95% CI [.22, .38]) and the second block (β = .20, p < .001, 95% CI [.12, .28]). Surprisingly, the frustration of efficacy presented a negative effect on populism in both the first (β = -.11, p = .004, 95% CI [-.19, -.04]) and second block (β = -.12, p = .004, 95% CI [-.20, -.05]). This may be due to strong collinearity with belonging (r = .57, p < .001); in fact, the bivariate correlation between the frustration of efficacy and populism was weak but positive (r = .10, p = .002).

GENERAL DISCUSSION

Hypotheses related to the main focus of these studies were confirmed. Participants perceiving themselves to occupy a low position in society were more likely to feel deprived compared to other social classes (intergroup), other individuals from the same social class (interindividual), and their own past position (temporal). These feelings of RD, as well as participants' perceived social status, were associated with holding an economic status-based identity that was experienced as psychologically threatening. This, in turn, predicted participants agreement with populist ideas.

In both studies, participants who felt a stronger sense of RD were more likely to report feelings of social exclusion (frustration of belonging) linked to their economic position in society, and these feelings of exclusion were linked to greater endorsement of populist thin ideology. Crucially, in Study 2 and replicating the results of Study 1, frustration of the belonging motive fully or partially mediated the pathways from all three forms of RD to populism. Further, our findings provided evidence concerning other predictors of populism outlined in previous literature. These results are discussed below and highlight how the roles of subjective economic indicators and identity threat (in particular social exclusion) might be interpreted within an economic distress model that predicts the adhesion to the populist thin ideology.

Identity Threat and Populist Thin Ideology

Investigating the role of identity threat as a direct predictor of the populist thin ideology was one of the main aims of our research. Although theoretical contributions have suggested investigating the role of identity processes [START_REF] Hogg | Self-uncertainty and populism: Why we endorse populist ideologies, identify with populist groups, and support populist leaders[END_REF][START_REF] Obradović | Understanding the psychological appeal of populism[END_REF], to the best of our knowledge there was not any empirical evidence on the implication of motivated identity construction nor any form of identity threat on the populist thin ideology.

The model that we corroborated suggests that starting from the perception of having a lower social position, individuals develop RD feelings which predict economic status-based identity threat which, in turn, predicts populism. Using the frustration of status-based identity motives as a measure of the identity threat allowed us to establish a missing link in the relationship between socialeconomic distress and populism.

Albeit novel, these findings are consistent with the theoretical literature which implicates identity processes in populism [START_REF] Hogg | Self-uncertainty and populism: Why we endorse populist ideologies, identify with populist groups, and support populist leaders[END_REF][START_REF] Obradović | Understanding the psychological appeal of populism[END_REF], as well as with empirical findings associating various perceptions of threat to major political attitudes and behaviors (Ben-Nun [START_REF] Bloom | The effect of perceived cultural and material threats on ethnic preferences in immigration attitudes[END_REF][START_REF] Jost | Ideological asymmetries and the essence of political psychology[END_REF].

These results might be interpreted as indicating an identity management strategy [START_REF] Blanz | Responding to negative social identity: A taxonomy of identity management strategies[END_REF] activated in response to the identity threat experienced from occupying a subordinate socialeconomic position: The perception of low socioeconomic status provokes sentiments of RD and identity threat, which, in turn, provoke the adhesion to the populist thin ideology providing a positive "good people" identity opposed to the negative outgroup "corrupt elite." Future research could investigate this further, for example by testing whether experimental manipulations of social exclusion lead people to increase their endorsement of populist ideology.

The Economic Distress Pattern and Its Integration With Identity Threat

Recent evidence from social and political sciences has highlighted the idea that economic distress fuels populism [START_REF] Oxendine | The political psychology of inequality and why it matters for populism[END_REF][START_REF] Rhodes-Purdy | Populist psychology: Economics, culture, and emotions[END_REF]. This hypothesis was confirmed in Study 1 and in Study 2. In particular, intergroup RD (Figures 2 and5) produced the highest effect size in Study 1 and was the only type of RD that was directly related to populism in Study 2. Further, intergroup RD was the strongest direct predictor of populism in both studies (Figures 2 and5). These findings fit well with the theoretical definition of populism given by [START_REF] Mudde | Populism: An ideational approach[END_REF], particularly with the first characteristicthat society is divided into a "good people" ingroup and a "corrupted élite" outgroup, in other words, the intergroup conflict between the people and the elite [START_REF] Obradović | Understanding the psychological appeal of populism[END_REF].

To the best of our knowledge, this was the first time the distinction between these three levels of RD was employed to investigate populism, and this helped us deepen our understanding of the phenomenon from an inner-attitudinal perspective [START_REF] Smith | Relative deprivation: A theoretical and metaanalytic review[END_REF]. In both studies, all three levels were directly or indirectly related to populism.

The distinction between interindividual and intergroup levels seemed particularly relevant because of their moderate correlation (Table S1 in the online supporting information) showing a substantial distinction of these two dimensions. This is also consistent with empirical evidence that integrates the concept of RD with social identity theory and suggests that intergroup and interindividual levels of deprivation should both be taken into account [START_REF] Abrams | Testing the social identity relative deprivation (SIRD) model of social change: The political rise of Scottish nationalism: Identity, deprivation, and voting intentions[END_REF][START_REF] Mummendey | Strategies to cope with negative social identity: Predictions by social identity theory and relative deprivation theory[END_REF].

Furthermore, all three levels of RD showed direct effects on the economic status-based identity threat, highlighting two important points: First, the employment of these three elements is not redundant but necessary to get a better understanding of some identity threat processes; second, the statusbased identity is both a social (collective) and an individual identity and incorporates a comparison with the past. The latter strengthens the assumption that this economic element might be perceived as both a collective and an individual identity element [START_REF] Vignoles | Identity: Personal and social[END_REF]. To conclude, the integration between economic distress pattern and identity threat as a predictive model of the populist thin ideology is an essential part of this framework.

The Role of Social Exclusion

In the present studies, the frustration of the belonging motive corresponded to the perception of social exclusion relative to the economic position occupied in society. Contrasting with the classic focus of social identity theory on positivity (i.e., self-esteem) and distinctiveness (e.g., [START_REF] Tajfel | An integrative theory of intergroup conflict[END_REF], our analyses showed that frustrated belonging was the best predictor of the populist thin ideology among identity motives. Moreover, this specific identity motive appeared to be central to this process: Firstly, when tested on the same data, the model with social exclusion (Figure 3) showed a very similar pattern to that with the global measure of identity threat (Figure 2). Second, the frustration of belonging was the only identity motive to show consistent and robust direct effects on the populist thin ideology across the two studies.

The role played by social exclusion here is consistent with findings that individuals who experience social exclusion tend to identify themselves with broader groups (e.g., the people instead of a specific social class; [START_REF] Knowles | Benefits of membership: The activation and amplification of group identities in response to social rejection[END_REF] and overestimate the dimensions of their ingroup [START_REF] Pickett | The impact of assimilation and differentiation needs on perceived group importance and judgments of ingroup size[END_REF]. This finding could also be related to the strategy to redefine ingroup borders [START_REF] Blanz | Responding to negative social identity: A taxonomy of identity management strategies[END_REF]. Future research could further investigate the implication of this particular identity motive within the economic distress pattern of populism.

Is the Cultural Backlash Pattern Also Related to Identity Threat?

Even though our main focus has been on predictors related to the economic distress pattern, we also included predictors linked to cultural backlash: national collective narcissism and anomie. The former was a direct predictor of populist thin ideology in both studies, whereas the latter did not show any effect on the populist thin ideology.

The lack of support for a direct effect of anomie contrasts with previous empirical evidence where anomie was a predictor of antielitist attitudes [START_REF] Hartwich | Exposure to neoliberalism increases resentment of the elite via feelings of anomie and negative psychological reactions: Neoliberalism and anti-elitism[END_REF]. The employed measure of anomie may not have caught all relevant aspects of the construct [START_REF] Elchardus | Populism, persistent republicanism and declinism: An empirical analysis of populism as a thin ideology[END_REF]: This measure focused particularly on the difficulty to understand the current world and less on the perception of erosion of society [START_REF] Teymoori | Revisiting the measurement of anomie[END_REF]. The role of anomie should be tested again using alternative measures. In contrast, the implication of national collective narcissism is coherent with studies showing a link between this construct and right-wing populism [START_REF] Marchlewska | Populism as identity politics: Perceived in-group disadvantage, collective narcissism, and support for populism[END_REF].

Together, these results suggest that the cultural backlash pattern is not an alternative to the economic distress pattern, but they act by parallel paths on the populist thin ideology. Correlations between these predictors and identity threat (Table S1 in the online supporting information) suggest that identity threat may be a mediator for the cultural pattern as well. However, appropriate examination of this question requires future research that measures threats to cultural aspects of identity, rather than to economic status-based identity.

Limitations and Future Directions

Future research should illuminate questions related to the limitations of these studies. A crosscultural approach could be adopted to test if these results are generalizable to other populations and contexts. Also, the cross-sectional design presents some limits: Correlations do not furnish a clear direction for causality, and the collinearity in regression models might, in some cases, yield artificial effects difficult to replicate. Our principal hypotheses were confirmed in both studies, and we did not find major problems of collinearity in these studies (except for the effect of efficacy mentioned in the results of Study 2), but experimental or longitudinal designs could help to provide more robust evidence for causal directions. Furthermore, a deeper focus on the role of social exclusion is recommended, especially keeping in mind the particular context in which the two studies were conducted. For some individuals, social distancing might have increased general feelings of social exclusion [START_REF] Graupmann | When ostracism is mandated: COVID-19, social distancing, and psychological needs[END_REF]. Even though belonging motive frustration in these studies was in relation to the economic status-based identity element, we cannot be certain whether social distancing affected this particular form of social exclusion, nor, more importantly, whether the relation between this identity motive and populism was affected. Nevertheless, the social distancing context (and more generally pandemic-related restrictions) varied significantly between the periods of data collection for the two studies. Similarity in results between Study 1 (strong lockdown) and Study 2 (flexible social distancing) suggests a minor role of social distancing on our results. Still, it would be useful to replicate these studies out of emergency context. Finally, the potential mediating role of forms of identity threat should be tested also for predictors relative to the cultural backlash postulate.

These studies support the thesis that identity motive frustration-particularly frustration of the belonging motive-is a psychological predictor of populism and plays a central role in the process by which economic deprivation triggers populism. This is the first empirical evidence to our knowledge highlighting the implication of identity threat as a predictor of populism. The framework that emerged implies a pattern that starts from the perception of low social status and, going through RD and identity threat, ends up with the behavioral outcome of a populist vote.

Figure 1 .

 1 Figure 1. Summary of the hypotheses in Study 1.

Figure 3 .

 3 Figure 3. Structural equation model (DWLS method) to explore results with a model including the frustration of belonging instead of the global latent factor of identity threat. This model includes social status deprivation, relative deprivation, identity threat, right (versus left) positions on economic and social issues, collective narcissism, social exclusion (frustration of belonging) and populist vote. N = 380. Participants who did not express their vote intention were excluded from these analyses. All regression weights and covariances were standardized. Fit indexes: (Χ 2 (df = 538, N = 380) = 1423.57, p < .001, Robust CFI = .902, Robust RMSEA = .075[.072, .078]). *p < .05, **p < .01 ***p < .001.

Figure 2 .

 2 Figure 2. Structural equation model (DWLS method) to test the hypotheses of Study 1, including social status deprivation, relative deprivation, identity threat, right (versus left) positions on economic and social issues, collective narcissism, populist thin ideology, and populist vote. N = 380. Participants who did not express their vote intention were excluded from these analyses. All regression weights and covariances were standardized. Fit indexes: (Χ 2 (df = 604, N = 380) = 1596.91, p < .001, Robust CFI = .904, Robust RMSEA = .075[.072, .077]). *p < .05, **p < .01 ***p < .001.

Figure 4 .

 4 Figure 4. Summary of the hypotheses in Study 2.

  Procedures and instruments were the same as in Study 1. Instruments were described under Study 1. The reliability indices computed on the Study 2 sample were: • Social exclusion/ frustration of belonging identity motive (Ω = .83) • Frustration of continuity (Ω = .75) • Frustration of distinctiveness (Ω = .64) • Frustration of efficacy (Ω = .79) • Frustration of self-esteem (Ω = .83) • Frustration of meaning (Ω = .78) • Intergroup RD (Ω = .87) • Interindividual RD (Ω = .83) • Temporal RD (Ω = .92) • National collective narcissism (Ω = .83) • Populist thin ideology (Ω = .89)

Figure 5 .

 5 Figure 5. Structural equation model (DWLS method) to test the hypotheses from Study 2. The model includes social status deprivation, relative deprivation, identity threat, right (versus left) positions on economic and social issues, collective narcissism, social exclusion (frustration of belonging), and populist vote. N = 860. Participants who did not express their vote intention were excluded from these analyses. All regression weights and covariances were standardized. Fit indexes: (Χ 2 (df = 538, N = 860) = 2004.19, p < .001, Robust CFI = .936, Robust RMSEA = .06. [.06, .06]). *p < .05, **p < .01 ***p < .001.

  .: Conceptualization, Methodology, Data Curation, Formal Analysis, Writing -Original Draft. M.B.: Conceptualization, Methodology, Supervision, Writing -Review & Editing. M.J.E.: Conceptualization, Methodology, Writing -Review & Editing. V.L.V.: Conceptualization, Methodology, Writing -Review & Editing.

Note that effects of perceived anomie and cultural narcissism on populism theoretically might also be mediated by identity threats. However, the focus of such threats would be individuals' sense of cultural identity, rather than their economic status in society. Since we did not measure cultural identity threat in the current studies, we treated these variables as direct predictors of populism in our hypotheses (Figure1).

ACKNOWLEDGEMENT

Correspondence concerning this article should be addressed to Efisio Manunta, CLLE, Maison de la recherche, 5 allées A. Machado, 31058 Toulouse cedex 9, France. E-mail: efisio.manunta@ univ-tlse2.fr

DATA ACCESSIBILITY STATEMENT

The preregistration for study 2 is openly available on the Open Science Framework (OSF) website at this link: "https://osf.io/fc9y7/" https://osf.io/myu7b?view_only=38b17fa9f3594da68c-c53ad32a49a527. The preregistration adheres to the disclosure requirements for the preregistered badge maintained by the Center for Open Science (https://osf.io/tvyxz/wiki/home/). Raw data and materials of both studies will be provided if requested by any scholar or scientific institution. This request must be addressed to the corresponding author.

Supporting Information

Additional supporting information may be found in the online version of this article at the publisher's web site:


Keywords: Sensitivity analysis, Virtual work method, Shape optimization, Electrical machine design

The sensitivity analysis of electromagnetic quantities such as the magnetic force and torque is a crucial step when optimizing electromagnetic devices with gradient-based optimizers. This paper provides an analytical derivation of those two quantities using the virtual work principle in a finite element package. It is suitable for any kind of optimization. A focus on the particular use of this sensitivity on shape optimization is presented. The development and validation of the method is detailed in this case. Finally, it is applied to the shape optimization of a switch reluctance motor (SRM) to improve its performances.

I. INTRODUCTION

T

The virtual work method [START_REF] Coulomb | A methodology for the determination of global electromechanical quantities from a finite element analysis and its application to the evaluation of magnetic forces, torques and stiffness[END_REF] is another method to compute the force as the derivative of the magnetic energy. It is known to have a better accuracy [START_REF] Ren | Comparison of different force calculation methods in 3D finite element modelling[END_REF]. Another advantage compared to the Maxwell tensor is that it does not require the additional choice of a path surrounding the region where the force is computed.

In [START_REF] Yoo | Optimal design of an electromagnetic coupler to maximize force to a specific direction[END_REF], the adjoint method was used on a virtual work formulation, but it uses the finite difference method. This paper provides a fully analytical sensitivity analysis of the magnetic force and torque via the virtual work method using the adjoint method. To our knowledge, this has never been done before.

Section II presents the magnetostatic context of the study, and the state of the art of the sensitivity analysis using the adjoint method. Section III presents the specific application of the adjoint method to the magnetic force and torque sensitivity computed using the virtual work method in a general optimization setup. Section IV details the specific application of section III to shape optimization and finally section V shows an example of its use on the optimization of a SRM built with a nonlinear ferromagnetic material.

II. STATE OF THE ART A. Magnetostatic context

Let Ω be the 2D domain of study. A current supply is represented using a density J = J z e z in a region Ω S and zero elsewhere. Let Ω F be a domain filled with a ferromagnetic material, possibly with nonlinear characteristics. The current J creates a magnetic force or torque on Ω F .

The magnetostatic equations with 2D assumptions satisfied in Ω are :

∇ × H = 0 in Ω\Ω S , ∇ × H = J in Ω S , (1)
H = νB ; B = ∇ × A z , A z = 0 on ∂Ω,
where H is the magnetic field, B is the magnetic induction, A z = A z e z is the 2D potential vector and ν is the magnetic reluctivity. In the air regions, one has ν = ν 0 with ν 0 the vaccum

Sensitivity Analysis Using the Virtual Work Principle for 2D Magnetostatic Problems

Olivier Brun 1,2 , Olivier Chadebec 1 , Pauline Ferrouillat 2 , Innocent Niyonzima 1 , Jonathan Siau 2 , Gérard Meunier 1 , Laurent Gerbaud 1 , Frédéric Vi 2 and Yann Le Floch 2 1 Univ. Grenoble Alpes, CNRS, Grenoble INP, G2ELab, Grenoble, France 2 Altair Engineering, Meylan,France HE optimization of electromagnetic devices is a popular topic. It allows to improve existing devices or to find new ones which may not be intuitive. Three types of optimization can be distinguished: parametric, shape and topology optimization. The main difference between those setups is the nature of the design space. In a parametric context, one mainly aims at optimizing predefined parameters [START_REF] Zhou | An integrated approach to topology, sizing, and shape optimization[END_REF], for instance geometric parameters. For shape optimization, it is the border nodes of an initial design which are considered [START_REF] Zhou | An integrated approach to topology, sizing, and shape optimization[END_REF], [START_REF] Kim | Applying continuum design sensitivity analysis combined with standard EM software to shape optimization in magnetostatic problems[END_REF]. In the case of topology optimization, the goal is to distribute a given amount of ferromagnetic material in the mesh elements of a chosen optimization region. The SIMP method [START_REF] Zhou | An integrated approach to topology, sizing, and shape optimization[END_REF] or the Level-set method [START_REF] Kim | Design optimisation of electromagnetic devices using continuum design sensitivity analysis combined with commercial EM software[END_REF] can be used.

Regardless of the optimization type, genetic algorithms can be numerically used. They converge towards the global optimum but are known to use unreasonable computation time when the number of optimization parameters becomes large. Another choice is the use of gradient-based algorithms. They converge faster but might be stuck in local optimums. A crucial step when using those algorithms is a precise computation of the sensitivity with respect to the design variables of the optimization. A basic finite difference method can be used, which is easy to implement but numerically slow and unstable. A widely used alternative is an analytical derivation of the optimization objective, coupled with the adjoint method in order to keep an acceptable computation time. The adjoint method is presented in many papers, for instance in [START_REF] Zhou | An integrated approach to topology, sizing, and shape optimization[END_REF], [START_REF] Kim | Applying continuum design sensitivity analysis combined with standard EM software to shape optimization in magnetostatic problems[END_REF] and [START_REF] Kim | Design optimisation of electromagnetic devices using continuum design sensitivity analysis combined with commercial EM software[END_REF].

A classical optimization for electromagnetic devices such as actuators or motors is to maximize the magnetic force or torque in their mobile part while constraining the volume of the device and possibly other electromagnetic quantities.

In [START_REF] Labbe | Topology optimization method based on the Maxwell stress tensor for the design of ferromagnetic parts in electromagnetic actuators[END_REF] and [START_REF] Lee | Isogeometric shape optimization of ferromagnetic materials in magnetic actuators[END_REF], the Maxwell tensor method is used to compute the force and its sensitivity using the adjoint method. reluctivity. In the regions filled with ferromagnetic materials, including Ω F , ν is described by nonlinear laws ν = ν(|B|).

This nonlinear scalar equation ( 1) is solved using the Newton-Raphson algorithm which iteratively solves the following matrix system :

S(A (k-1) )A (k) = b (k-1) (2)
where A (k) is the vector of values of the potential vector A z discretized on the mesh nodes of the domain Ω at the k th current iteration of the Newton-Raphson algorithm.

B. The adjoint method

Let F be a scalar function of interest that has to be optimized and (X i ) 1≤i≤n some design variables in a general optimization context. F is a function of the design variables and of the state variable:

F = F (A, X 1 , . . . , X n ) with A = (A j ) j=1,.
.,m the nodal values of the vector potential. Assuming that the Newton-Raphson algorithm (2) converges in K+1 iterations, the state variable A is obtained by solving a linear system S (K) A = b (K) . The sensitivity using the adjoint method reads [START_REF] Zhou | An integrated approach to topology, sizing, and shape optimization[END_REF]:

dF dX i = ∂F ∂X i + λ T db (K) dX i - dS (K) dX i A . (3)
with λ the adjoint vector defined as the solution of :

S (K) λ = ∂F ∂A = ∂F ∂A i 1≤i≤m . (4)
The strength of this method is that λ is the solution of a linear matrix system already built during the physical solving [START_REF] Kim | Applying continuum design sensitivity analysis combined with standard EM software to shape optimization in magnetostatic problems[END_REF]. The terms db (K) dXi and dS (K) dXi can be analytically expressed from the finite element assembly.

III. SENSITIVITY ANALYSIS WITH THE VIRTUAL WORK METHOD

A. The virtual work method for magnetic force and torque

The virtual work method allows to compute the magnetic force on Ω F as the derivative of the magnetic energy W = Ω B 0 H(b)dbdΩ along a direction s at constant state variable A. After breaking down the integral over the mesh, its expression is [START_REF] Coulomb | A methodology for the determination of global electromechanical quantities from a finite element analysis and its application to the evaluation of magnetic forces, torques and stiffness[END_REF]:

F s = - ∂W ∂s = ν 0 e Ωe B T M e B dΩ (5)
with

M e = -G -1 e ∂G e ∂s + 1 2|G e | ∂|G e | ∂s I,
where G e is the Jacobian matrix of the transformation from the mesh element Ω e to the reference element ∆ e [START_REF] Coulomb | A methodology for the determination of global electromechanical quantities from a finite element analysis and its application to the evaluation of magnetic forces, torques and stiffness[END_REF], |G e | its determinant and I is the identity matrix. The terms ∂Ge ∂s and ∂|Ge| ∂s are the derivatives of G e and |G e | with respect to the translation of the mesh nodes of Ω F in the direction s. They are zero everywhere except in the first air layer of mesh elements surrounding Ω F .

The same formula remains true for the torque around an axis ω, but in this case the mesh nodes of Ω F are moved with a rotating movement around ω.

B. Expression of the sensitivity

As the adjoint vector λ is the solution of the linear problem (4), the term ∂Fs ∂A has to be expressed in the case of the force [START_REF] Lee | Isogeometric shape optimization of ferromagnetic materials in magnetic actuators[END_REF]. One has:

∂F s ∂A = ∂F s ∂B T ∂B ∂A . (6)
Since B = m j=1 A j ∇ × τ j with τ j the 2D nodal shape function at node j, the vector ∂B ∂A can be immediately written as :

∂B ∂A 1≤j≤m = (∇ × τ j ) 1≤j≤m . (7)
The magnetic force has an explicit formula in term of B field (5) that can be directly differentiated.

∂F s ∂B = ν 0 e Ωe (M e + M T e )B dΩ. (8)
Combining the two last equations, it is possible to build ∂Fs ∂A and to solve the adjoint equation ( 4). The adjoint state is the same regardless of the chosen type of optimization. Then, the sensitivity can be computed using (3). However, the term ∂Fs ∂Xi in (3) depends on the considered optimization type. Since the support of the magnetic force is the first air mesh layer surrounding Ω F , ∂Fs ∂Xi is non zero only when the parameters (X i ) 1≤i≤n influence the mesh elements of this layer. For instance, when the SIMP method is used in the case of topology optimization, ∂Fs ∂Xi is always zero. The next section details the case of shape optimization, where ∂Fs ∂Xi can be non-zero.

IV. SPECIFIC CASE OF SHAPE OPTIMIZATION A. Explicit dependency of the force on the design variables

In the case of shape optimization, the design variables are the nodes (X i ) 1≤i≤n of the border lines of an initial design. The adjoint state can be solved using ( 4) and [START_REF] Coulomb | A methodology for the determination of global electromechanical quantities from a finite element analysis and its application to the evaluation of magnetic forces, torques and stiffness[END_REF]. The term ∂Fs ∂Xi in (3) is zero in general except when optimizing the border lines of Ω F . In this case, the optimization is performed at the same place where the force is computed. It means that there is an explicit dependency of the force on the X i variables and the term ∂Fs ∂Xi is non-zero. It must be taken into account but its development and implementation is tedious.

The term has been expressed by applying twice the virtual work principle on the magnetic energy W , which leads to : with respect to the virtual translation of the node X i in the direction of computation of the sensitivity. They are zero everywhere except on the elements that contain X i . This formula is deduced using the specificity of the potential vector formulation. Indeed, the transformation to the reference space requires the use of a Piola transform which is not the same for scalar and vector formulations [START_REF] Zaglmayr | High Order Finite Element Methods for Electromagnetic Field Computation[END_REF]. Using a scalar potential formulation H = -∇ψ would lead to a different formula. See [START_REF] Coulomb | A methodology for the determination of global electromechanical quantities from a finite element analysis and its application to the evaluation of magnetic forces, torques and stiffness[END_REF] for a detailed use of the scalar formulation on the virtual work method.

∂F s ∂X i = - ∂ 2 W ∂s∂X i = N e=1 ∆e ∂ 2 |G e | ∂s∂X i H 0 BdH (9)

B. Validation

The sensitivity (3) of the magnetic force and torque has been implemented in the case of shape optimization, including ( 6) and the tedious term [START_REF] Zaglmayr | High Order Finite Element Methods for Electromagnetic Field Computation[END_REF]. A validation step has been performed on a simple bidimensional academic test case using the following (central) finite difference method to confirm the implementation :

dF s dX i ≈ F s (X i + ϵ) -F s (X i -ϵ) 2ϵ . ( 10
)
In the context of shape optimization, X i + ϵ stands for a slight disturbance of the coordinates of X i either in the horizontal or the vertical direction, depending on the direction of computation of the sensitivity. The geometry of the test case is presented in Figure 1. A current supply is represented by the orange coil. A region Ω F is filled with a nonlinear ferromagnetic material. The sensitivities of the force and torque on Ω F are calculated on the n = 77 mesh nodes of Ω F (thick lines on figure 1). The finite difference method is really slow since it requires 2n nonlinear finite element solvings per sensitivity computation direction. In contract, the adjoint method (3) is much faster because it only needs the additional solving of (4). Figures 2 and3 compare the sensitivity values obtained with the two methods. The maximum relative error is lower than 1%. The use of the Maxwell tensor instead of the virtual work method would lead to a similar computational cost.

V. APPLICATION A. Optimization workflow

After the concluding validation step, the sensitivity code has been included in a gradient-based shape optimization workflow using the softwares Altair Flux TM and Optistruct TM . Starting from a given design, the sensitivity is used to update the current optimization nodes in a descent direction. Then, the process iterates until the optimization converges to an optimal design or a maximum number of iteration is reached. Fig. 3: Sensitivity with respect to a vertical displacement of the nodes obtained with the finite difference method (FD) and the adjoint method (AM)

B. Optimization problem

This optimization algorithm is applied on a realistic SRM model presented in figure 4. The stator (in orange) and rotor (in grey) are made of a laminated nonlinear ferromagnetic material. The current supply in the stator has three winding phases represented in red, blue, green and brown. Thanks to the geometrical and electrical symetries, only one quarter of the motor can be represented. The light blue color represents the air regions of the device.

The torque depends on the position of the rotor. Thus, the angular position of the rotor is discretized in N = 30 positions (θ i ) 1≤i≤N to compute it in every rotor configuration.

The main objective is to maximize the mean torque value T mean over all the rotor positions :

T mean = 1 N N i=1 T (θ i ), (11)
with T (θ i ) the torque computed at the angular position θ i . The SRM are known to have a large torque amplitude over a rotation. This leads to noisy motors. A relative measure of this amplitude is given by the torque ripple T r :

T r = 100 × T max -T min T mean , (12)
where

T min = min 1≤i≤N T (θ i ) and T max = max 1≤i≤N T (θ i ).
The torque ripple must remain under an acceptable value T max r in the optimized design. The sensitivity of the A constraint on the rotor volume V is also enforced to reduce by 20% the initial volume V 0 of the rotor represented in figure 4 and avoid the use of the expensive ferromagnetic material. The design variables of the shape optimization are the nodes (X i ) 1≤i≤886 of the lines inside the rotor in contact with the air regions. The final optimization problem is the following :

maximize X i T mean (X i ) subject to T r (X i ) ≤ T max r , V ≤ 0.8 × V 0 (13)

C. Results

After thirty optimization iterations and about two hours of computation, the optimized design shown in figure 5 has been obtained. The final design has a mean torque 38% higher than the initial one, while the mass of its rotor has decreased by 20% and its torque ripple decreased by 31%. Figure 6 shows the torque as a function of the angular position of the rotor for the initial and final designs. The optimized electrotechnical features are really interesting, however the final design clearly lacks of mechanical strength. It could be adressed by adding structural constraints to the optimization problem (13).

VI. CONCLUSION

The sensitivity analysis of the magnetic force and torque is analytically derived using the virtual work method. The expression of the sensitivity relies on the well known adjoint method, but it had never been applied to the virtual work method before. The sensitivity expression is presented in a general optimization context. Then, it is applied to the specific case of shape optimization. Finally, the use of this sensitivity analysis on the shape optimization of a SRM shows an interesting final design with an improved torque and reduced volume and torque ripple. An application of this method in the 3D case could be studied with a cautious adaptation of the explicit dependency (9) depending on the formulation (vector or scalar) which is used.

Fig. 2 :

 2 Fig.2: Sensitivity with respect to an horizontal displacement of the nodes obtained with the finite difference method (FD) and the adjoint method (AM)

Fig. 4 :

 4 Fig. 4: View of the initial SRM rotor design

Fig. 5 :Fig. 6 :

 56 Fig. 5: View of the optimized SRM rotor design

  Ge ∂s∂Xi and ∂ 2 |Ge| ∂s∂Xi are respectively the derivatives of the previous terms G e , |G e |, ∂Ge ∂s and

	-B T H + B T ∂G e ∂|G e | ∂s ∂X i T G -T ∂|G e | ∂X i e H ∂|G e | 1 |G e | ∂s + B T G -1 e ∂G e ∂s H ∂|G e | ∂X -J +J Ω F Fig. 1: Geometry of the academic test case where the terms ∂Ge ∂Xi , ∂|Ge| ∂Xi , ∂ 2 ∂|Ge| ∂s

i -B T G -1 e ∂ 2 G e ∂s∂X i H|G e | -B T G -1 e ∂G e ∂s ∂G e ∂X i T G -T e H|G e | d∆ e ,