Raoul Gucek
email: raoulgucek@gmail.com

David Le Gac
email: david.le.gac@gmail.com

PROSODIC MANIFESTATIONS OF POLITENESS IN PORTEÑO SPANISH WH-INTERROGATIVES: TERMINAL CONTOURS, F0 MITIGATION AND SYLLABLE DURATIONS

Keywords: wh-interrogatives, politeness, prosodic mitigation, terminal contour, Porteño Spanish

 regarding whinterrogatives' pitch contours were attributed to a mark of politeness.

The analysis of 280 wh-interrogatives produced by nine speakers of Porteño Spanish, within colloquial and polite contexts, allowed us to determine that politeness does not actually favour higher F0. On the contrary, politeness is expressed by F0 mitigation through lowered overall pitch and reduced span. Our results also refute [26]'s hypothesis that interrogative adverbs would make utterances containing them more sensitive to politeness. In Porteño Spanish, adverbs do not pilot rising terminal contours to signal politeness. As for syllable durations, lengthening did not occur and speech rate kept constant in our corpus in the polite condition.

INTRODUCTION

In Spanish, the intonation of wh-interrogatives has been described as roughly corresponding to that of declaratives [START_REF] Alcoba | Intonation in Spanish[END_REF][START_REF] Enbe | Sociolinguistic variation in the prosody of Buenos Aires Spanish according to the theory of phonology as human behaviour[END_REF][START_REF] Quilis | Tratado de fonología y fonética española[END_REF][START_REF] Sosa | La entonación del español. Su estructura fónica, variabilidad y dialectología[END_REF]. More specifically, [START_REF] Barjam | The intonational phonology of Porteño Spanish[END_REF] and [START_REF] Gabriel | Argentinian Spanish intonation[END_REF] explain that wh-words typically start the utterance carrying a high or rising accent (e.g. H* or L+H* in the ToBI system [START_REF] Vilaplana | La notación prosódica en español. Una revisión del Sp_ToBI[END_REF]), and the end of the interrogative is characterized by a progressive falling contour with the nuclear accent L* followed by an L% boundary tone. Yet, a rising final contour is reported by [START_REF] Navarro Tomás | Manual de entonación española[END_REF][START_REF] Quilis | Tratado de fonología y fonética española[END_REF] to soften the interrogation and make it sound polite. Recent studies on Porteño Spanish prosody [START_REF] Barjam | The intonational phonology of Porteño Spanish[END_REF][START_REF] Enbe | Sociolinguistic variation in the prosody of Buenos Aires Spanish according to the theory of phonology as human behaviour[END_REF][START_REF] Gabriel | Argentinian Spanish intonation[END_REF] have obtained contradictory results regarding wh-interrogatives' terminal contours. [START_REF] Barjam | The intonational phonology of Porteño Spanish[END_REF] and [START_REF] Gabriel | Argentinian Spanish intonation[END_REF] have found that falling terminal contours were the most commonly used, whereas for [START_REF] Enbe | Sociolinguistic variation in the prosody of Buenos Aires Spanish according to the theory of phonology as human behaviour[END_REF] it is the rising ones. [START_REF] Gabriel | Argentinian Spanish intonation[END_REF] suggested that H% might be predominant in oxytonic final words but also made various suggestions to explain these discrepancies: the lack of context in [START_REF] Enbe | Sociolinguistic variation in the prosody of Buenos Aires Spanish according to the theory of phonology as human behaviour[END_REF]'s study, different pragmatic goals (such as information seeking, inviting, criticizing, etc.) or the absence of control of sociolinguistic variables in all of the three studies. [START_REF] Enbe | Sociolinguistic variation in the prosody of Buenos Aires Spanish according to the theory of phonology as human behaviour[END_REF] also point out differences in the tasks proposed to the speakers: repetition of utterances in their study, reading out loud for [START_REF] Barjam | The intonational phonology of Porteño Spanish[END_REF] and semispontaneous speech for [START_REF] Gabriel | Argentinian Spanish intonation[END_REF]. The latter is recognized by [START_REF] Hübscher | Prosodic mitigation characterizes Catalan formal speech: The Frequency Code reassessed[END_REF][START_REF] Velásquez Upegui | La entonación de enunciados interrogativos transaccionales en el español hablado en Colombia[END_REF] as possibly enhancing intonation and making it more discriminant among contexts.

In her study of the prosody of wh-interrogatives in Colombian Spanish, [START_REF] Velásquez Upegui | La entonación de enunciados interrogativos transaccionales en el español hablado en Colombia[END_REF] also considers rising terminal contours as polite and further states that the grammatical category of the wh-word could be of influence. Specifically, adverbs seem to be capable of making the interrogatives containing them more sensitive to colloquial than polite contexts.

As for politeness, the separation line with formal speech is blurry and both terms are frequently employed as synonyms. [START_REF] Ide | Japanese sociolinguistics politeness and women's language[END_REF] connects these notions, stating that the use of formal forms creates a formal atmosphere where participants, being kept away from each other, avoid imposition. Thus, to create this formal atmosphere is to be polite. This conjugated effort of preservation corresponds to the facework [START_REF] Brown | Politeness: Some universals in language use[END_REF][START_REF] Goffman | La présentation de soi, 2. Les relations en public[END_REF] that participants put into place in face-threatening acts. A way to achieve thisapart from the use of adapted vocabulary and syntax, for exampleis through prosodic modulations, whose origins could be biological. The Frequency Code [START_REF] Gussenhoven | The phonology of tone and intonation[END_REF][START_REF] Ohala | Cross-language use of pitch: An ethological view[END_REF] believes that the prosodic colloquial vs polite forms are based on a morphological-acoustic correlate: the bigger the larynx, the lower the pitch. This leads to a universal biological code, which associates high frequencies with submission and low frequencies with domination. [START_REF] Gussenhoven | The phonology of tone and intonation[END_REF] then extrapolates from this a paralinguistic meaning with an affective interpretation of these parameters: namely, friendliness and politeness are associated with 'submissiveness' and a higher pitch. [START_REF] Hübscher | Prosodic mitigation characterizes Catalan formal speech: The Frequency Code reassessed[END_REF] compiled studies in various languages in favour of the Frequency Code (like [START_REF] Herraiz | The intonation of mitigating politeness in Catalan[END_REF] in Catalan, [START_REF] Hidalgo | Aprender a ser cortés: Aspectos pragmáticos de la enseñanza de ELE, expresividad fónica y comunicación cortés a través de la entonación[END_REF] in Peninsular Spanish and [START_REF] Tsuji | The case study of high pitch register in English and in Japanese: Does high pitch register relate to politeness?[END_REF] in Japanese). However, they also list many others against it. For instance, [START_REF] Grawunder | Politeness, culture, and speaking task -Paralinguistic prosodic behavior of speakers from Austria and Germany[END_REF] found that despite individual differences among speakers, average F0 measures seem to be lower in the formal register condition in German. In Korean as well, low pitch rather than high pitch was used to mark a polite register when addressing a person of superior status or age [START_REF] Winter | The phonetic profile of Korean formal and informal speech registers[END_REF]. [START_REF] Hübscher | Prosodic mitigation characterizes Catalan formal speech: The Frequency Code reassessed[END_REF] and [START_REF] Hübscher | Preschoolers use prosodic mitigation strategies to encode polite stance[END_REF]'s studies in Catalan also show F0 mitigation (i.e. the lowering of the overall pitch) happening in polite speech.

As far as the other prosodic cues are concerned, it is reported that politeness is expressed in particular by a slower speech rate. This phenomenon has been consistently described [START_REF] Grawunder | Politeness, culture, and speaking task -Paralinguistic prosodic behavior of speakers from Austria and Germany[END_REF][START_REF] Gryllia | The role of pragmatics and politeness in explaining prosodic variability[END_REF][START_REF] Hübscher | Prosodic mitigation characterizes Catalan formal speech: The Frequency Code reassessed[END_REF][START_REF] Hübscher | Preschoolers use prosodic mitigation strategies to encode polite stance[END_REF][START_REF] Lin | An acoustic study on the paralinguistic prosody in the politeness talk in Taiwan Mandarin[END_REF][START_REF] Ofuka | Prosodic cues for rated politeness in Japanese speech[END_REF][START_REF] Santabalbina | Duración y percepción de la cortesía en español (experimento piloto)[END_REF][START_REF] Winter | The phonetic profile of Korean formal and informal speech registers[END_REF] with no contradictory result so far.

In this paper, we present the results of a pilot study on the prosodic expressions of politeness in wh-interrogatives in Porteño Spanish. In particular, we aim to determine whether, in accordance with the Frequency Code, the polite context favours the rising pitch contours at the end of wh-interrogatives as suggested [START_REF] Enbe | Sociolinguistic variation in the prosody of Buenos Aires Spanish according to the theory of phonology as human behaviour[END_REF][START_REF] Navarro Tomás | Manual de entonación española[END_REF][START_REF] Quilis | Tratado de fonología y fonética española[END_REF][START_REF] Velásquez Upegui | La entonación de enunciados interrogativos transaccionales en el español hablado en Colombia[END_REF] or whether the wh-word category [START_REF] Velásquez Upegui | La entonación de enunciados interrogativos transaccionales en el español hablado en Colombia[END_REF] and the stress pattern of the final word [START_REF] Gabriel | Argentinian Spanish intonation[END_REF] play a role in the selection of the terminal contour. In addition to these local properties, we also investigate overall F0 and temporal characteristics of the wh-interrogatives such as the average pitch, the pitch range and syllable duration.

METHODOLOGY

Participants

Nine speakers of Porteño Spanish (5 females, 4 males; mean age 42.1, minimum 20, maximum 58) were recorded. They were all educated, either through academic studies or professional training.

Materials

The materials consisted in a corpus of 32 whinterrogatives equally divided between a colloquial and a polite context, with each of these employing the same number of adverbs and pronouns. According to the Diccionario de la Real Academia Española [START_REF]Diccionario de la lengua española[END_REF], interrogative adverbs are cómo (how), cuándo (when), dónde (where) and por qué (why), and interrogative pronouns are cuál (which), cuánto (how much/long), qué (what) and quién (who). As for the stress pattern of the final words, the balance of (non-)oxytonic final words was roughly the same for each wh-word and formality condition.

The utterances were contextualized in order to carry a colloquial or polite value. The examples in (1) and ( 2) show discourse contexts encoding a colloquial situation and its polite correspondent, respectively, to elicit the target sentence (TS) ¿Cuál me queda mejor? (Which one suits me better?).

(1) Te vas de fiesta y no sabes qué chaqueta ponerte, dudas entre dos. Se las enseñas a un amigo y le preguntas: TS (You're going to a party and you're thinking about which jacket to put on, you're wavering between two. You show both of them to a friend and ask him: TS)

(2) Está en una tienda probándose trajes/vestidos para un casamiento. Le pregunta al vendedor: TS (You are in a shop trying on a dress/suit for a wedding. You ask the shop assistant: TS) Two of the three pragmatic factors of [START_REF] Hübscher | Prosodic mitigation characterizes Catalan formal speech: The Frequency Code reassessed[END_REF] were implemented in the discourse contexts presented to the participants: the social distance between the interlocutors and their relative power. The third one, the cost of the action, does not present any opportunity to be tested using wh-interrogatives.

Experimental procedure

The speakers were recorded in Buenos Aires, Argentina. The recordings took place in quiet rooms with a Zoom H1n microphone, in stereo at 24-bit/48 kHz and with a saturation limiter activated.

Each speaker was presented with plasticized cards in different randomized orders, containing a TS introduced by the description of the context. To avoid boredom and fatigue, only half of the corpus was offered to each of them. This represented one of four TSs introduced by an adverb and one of four of those introduced by a pronoun, induced in the colloquial contextwhich was always offered firstand then their eight polite correspondents. Participants were invited to ask any question that would be necessary to clarify the contextualization. The readings were systematically repeated once in order to discard possible faulty ones. In the end, 280 utterances were exploitable.

Labelling, measurements and quantitative analysis

The TSs were segmented and labelled using Praat software [START_REF] Boersma | Praat: Doing phonetics by computer[END_REF]. They were manually segmented into syllables and an F0 measurement was taken at times ⅓, ½ and ⅔ of the duration of each syllable. From these measurements, the following data were established for each entire wh-interrogative/TS: 1. average pitch, min and max F0 values in Hz; 2. F0 value on the middle of the stressed syllable of the wh-words; 3. F0 value at the end of the TSs, which represents the F0 target of the final boundary tone of the sentence;

4. pitch range in semitones (ST) defined as the difference between the max and min F0 of the TSs (the min F0 was taken as the reference value); 5. syllable duration of the wh-words and of the last two syllables of the TSs; 6. speaking rate over the entire TSs defined as the duration of the interrogation divided by the number of syllables.

As for the pitch contours, they were annotated in an independent tier following [START_REF] Vilaplana | La notación prosódica en español. Una revisión del Sp_ToBI[END_REF]'s Sp_ToBI annotation system, as shown in Fig. 1:

RESULTS

Nuclear pitch configurations

Each nuclear pitch configuration (NPC), i.e. the nuclear pitch accent plus boundary tone(s) for each wh-interrogative, was annotated. Table 1 below presents the distribution of the NPCs in the two formality conditions. It also includes the grammatical category of the wh-word to establish this distribution. The distribution of the NPCs is similar in colloquial and polite contexts. More specifically, L* L% is the most widely used contour in both formality conditions (63.9% in the colloquial condition and 72.1% in the polite one). Slightly more L* L% occurred in the polite context but this difference cannot be considered significant. Overall, the falling NPCs (i.e. those consisting in a pitch drop at the end of the sentence) clearly outnumber the rising ones, which end with a slight (L* M%) or a steep F0 rise ((H)H%): there is 73.6% of falling NPCs vs 26.4% of rising NPCs in the colloquial condition and 79.4% vs 20.6% of each contour type in the polite one. The results thus indicate that the politeness condition does not favour a terminal rising pitch.

The grammatical category of the wh-word does not have any influence either over NPCs. Table 1 shows that interrogative adverbs and pronouns generate equivalent quantities of rising contours in both formality conditions.

With regard to [START_REF] Gabriel | Argentinian Spanish intonation[END_REF]'s (p. 290) claim that H% boundary tone typically occurs in questions ending in oxytonic words, the overall results do not seem to support this finding. As can be seen in Fig. 2, the distribution of the boundary tones is quite similar at the end of the oxytonic and non-oxytonic words. Nevertheless, if one puts aside the predominance of L% and considers the cases where speakers have not chosen the L% tone, there seems to be a preference for the (H)H% categories at the end of the oxytonic words (17.6%/22.9%=76.9%), while the M% seems to be preferred with non-oxytonic words (15.7%/23.6%=66.5%). However, it is not yet possible to conclude because of insufficient data. (t-tests). For these measurements and mean comparisons, we have selected only TSs that were strictly identical (indeed, some TSs had to be formulated differently according to the context to ensure coherence and authenticity). This amounted to 134 utterances in each context. As can be seen, in the polite condition, all F0 values (average pitch, min, max, Inter. and Fin.) are significantly lower (p<0.001, p=0.004 for Fin.) than those in the colloquial condition. This suggests that F0 mitigation in the polite context affects the whole wh-interrogative and not a specific contour or word. The only non-significant difference between contexts is the pitch range, which corroborates the idea that it is the overall F0 pattern of the sentences that is lowered in the polite context.

Overall pitch characteristics of wh-interrogatives

As regards the location of the F0 maxima, [START_REF] Sosa | La entonación del español. Su estructura fónica, variabilidad y dialectología[END_REF] (p. 216) pointed out that the highest peak tends to coincide with the wh-word (interrogative pronoun or adverb), although not necessarily with its accented syllable, without dialectal distinctions. In Porteño Spanish, the results show that the F0 max is not located on the stressed syllable of the wh-word in most cases. This may well be a case of late tonal alignment (as in Fig. 1 with L+>H*), which will require further investigation.

Syllable durations

Table 3 presents the mean duration of the last two syllables of the wh-word and of the TSs in the two formality conditions. It also gives the speaking rate of the TSs (i.e. duration of the TS divided by the number of syllables, cf. 2.4.).

Unlike the findings of [START_REF] Grawunder | Politeness, culture, and speaking task -Paralinguistic prosodic behavior of speakers from Austria and Germany[END_REF][START_REF] Gryllia | The role of pragmatics and politeness in explaining prosodic variability[END_REF][START_REF] Hübscher | Prosodic mitigation characterizes Catalan formal speech: The Frequency Code reassessed[END_REF][START_REF] Hübscher | Preschoolers use prosodic mitigation strategies to encode polite stance[END_REF][START_REF] Lin | An acoustic study on the paralinguistic prosody in the politeness talk in Taiwan Mandarin[END_REF][START_REF] Ofuka | Prosodic cues for rated politeness in Japanese speech[END_REF][START_REF] Santabalbina | Duración y percepción de la cortesía en español (experimento piloto)[END_REF][START_REF] Winter | The phonetic profile of Korean formal and informal speech registers[END_REF], Table 3 shows that politeness does not affect either syllable duration or speech rate in Porteño Spanish. No significant differences can be seen for the mean durations between the two formality conditions.

Table 3: Mean duration in seconds of the wh-word ("Inter."), the last two syllables of the TSs ("Fin.") and the syllables over all the TSs (representing the speaking rate) in the two formality conditions. The table also includes the results of a series of t-tests

DISCUSSION AND CONCLUSION

In this paper we aimed to determine what the prosodic manifestations of politeness are in whinterrogatives in Porteño Spanish. Our results show that politeness has no influence on NPCs. Indeed, a final L% occurs in the majority of the interrogatives (more than 75%) no matter the formality condition. This means that politeness does not favour high or rising NPCs, contrary to [START_REF] Enbe | Sociolinguistic variation in the prosody of Buenos Aires Spanish according to the theory of phonology as human behaviour[END_REF][START_REF] Navarro Tomás | Manual de entonación española[END_REF][START_REF] Quilis | Tratado de fonología y fonética española[END_REF]'s proposals. The grammatical category of the wh-word has no influence either on the NPCs, in contrast to [START_REF] Velásquez Upegui | La entonación de enunciados interrogativos transaccionales en el español hablado en Colombia[END_REF]'s allegation in Colombian Spanish; neither does the stress pattern of the final word as claimed by [START_REF] Gabriel | Argentinian Spanish intonation[END_REF].

As for global F0 values (average pitch, min and max F0, Inter. and Fin.), mitigation occurs in all of these parameters (except for the pitch range) in the polite condition. Our results thus contradict previous research and in particular the Frequency Code premises, which predict that high or rising F0 is a cue for politeness. They are in line with the studies of [START_REF] Hübscher | Prosodic mitigation characterizes Catalan formal speech: The Frequency Code reassessed[END_REF][START_REF] Hübscher | Preschoolers use prosodic mitigation strategies to encode polite stance[END_REF] for Catalan, [START_REF] Grawunder | Politeness, culture, and speaking task -Paralinguistic prosodic behavior of speakers from Austria and Germany[END_REF] for German and [START_REF] Winter | The phonetic profile of Korean formal and informal speech registers[END_REF] for Korean, who also found that F0 mitigation occurred in the polite condition.

Another robust prosodic cue for politeness reported in earlier studies [START_REF] Grawunder | Politeness, culture, and speaking task -Paralinguistic prosodic behavior of speakers from Austria and Germany[END_REF][START_REF] Gryllia | The role of pragmatics and politeness in explaining prosodic variability[END_REF][START_REF] Hübscher | Prosodic mitigation characterizes Catalan formal speech: The Frequency Code reassessed[END_REF][START_REF] Hübscher | Preschoolers use prosodic mitigation strategies to encode polite stance[END_REF][START_REF] Lin | An acoustic study on the paralinguistic prosody in the politeness talk in Taiwan Mandarin[END_REF][START_REF] Ofuka | Prosodic cues for rated politeness in Japanese speech[END_REF][START_REF] Santabalbina | Duración y percepción de la cortesía en español (experimento piloto)[END_REF][START_REF] Winter | The phonetic profile of Korean formal and informal speech registers[END_REF] is a slower speech rate. Unexpectedly, we did not find this tendency in our data. The duration of the wh-words and that of the last syllables of the sentences exhibit no significant difference between the formality conditions. Likewise the speech rate remains the same in both contexts.

Further research on wh-interrogatives' prosody in Porteño Spanish still needs to work out how NPCs are chosen between rising and falling. The location of the F0 max is of interest too, as it appears that the F0 max is not aligned with the stressed syllable of the wh-word in Porteño Spanish. Finally, and as proposed by [START_REF] Hübscher | Prosodic mitigation characterizes Catalan formal speech: The Frequency Code reassessed[END_REF][START_REF] Velásquez Upegui | La entonación de enunciados interrogativos transaccionales en el español hablado en Colombia[END_REF], the present results would benefit from being tested with a different elicitation task, such as analysing (semi-) spontaneous speech.

ACKNOWLEDGEMENTS

We thank the participants of the study and David Imbert (CNRS LLING, Université de Nantes) for his help and advice with the statistical tests; all remaining errors are ours.

Figure 1 :

 1 Figure 1: Illustration of annotations for the TS ¿Dónde dejo el vino? (Where should I leave the wine?), in colloquial context and introduced by an adverb.

Figure 2 :

 2 Figure 2: Distribution of the boundary tones depending on the (non-)oxytonicity of the final word.

Table 1 :

 1 Inventory of NPCs within colloquial and formal contexts in % and actual number; "Adverbs" and "Pronouns" indicate the category of the wh-word.

Table 2

 2 below presents the mean F0 and standard deviation (SD) for the six pitch values defined in 2.4. in the two formality conditions. The table also contains the results of a series of mean comparisons

							COLLOQUIAL					POLITE
			Adverbs	Pronouns	Total	Adverbs	Pronouns		Total
			% N	% N	% N	% N	% N	% N
	Falling NPCs	L* L% H+L* L% other L%s H* M%	29,2% 42 2,8% 4 2,8% 4 1,4% 2	34,7% 50 0,0% 0 2,1% 3 0,7% 1	63,9% 92 2,8% 4 4,9% 7 2,1% 3	34,6% 47 2,2% 3 3,7% 5 0,0% 0	37,5% 51 1,5% 2 0,0% 0 0,0% 0	72,1% 98 3,7% 5 3,7% 5 0,0% 0
	Rising NPCs	L* M% L* (L)H% L+H* H% L+H* HH%	4,2% 1,4% 5,6% 2,8%	6 2 8 4	5,6% 0,0% 3,5% 3,5%	8 0 5 5	9,7% 14 1,4% 2 9,0% 13 6,3% 9	3,7% 2,9% 2,9% 2,9%	5 4 4 4	1,5% 1,5% 2,9% 2,2%	2 2 4 3	5,1% 4,4% 5,9% 5,1%	7 6 8 7
		Total					100,0% 144					100,0% 136

Table 2 :

 2 Mean F0 and standard deviation (SD) for the 6 pitch values defined in 2.4. in the two formality conditions. "Inter." and "Fin." refer to the F0 value on the wh-word and to the final boundary tone of the TSs, respectively. The table also includes the results of a series of t-tests (the degrees of freedom are 133 in all cases)


Arthur Guezengar

Keywords: Arendt, Hannah (1906-1975), Political philosophy, Imperialism, Slavery

Hannah Arendt's early work focused on the relationship between political violence, race, and nation. Arendt is one of the first thinkers to have emphasized the link between imperialism and totalitarianism. As soon as the Origins of Totalitarianism appeared, it underscored the connection between colonial imperialism and totalitarianism. The concepts of race and bureaucracy, which are at the heart of the totalitarian process, emerge from the colonial experience. In the same way the tools of the domination of the Western peoples over the colonized peoples, were the matrix from which the totalitarian domination could be concretized. Moreover, Hannah Arendt theorizes an opposition between power, violence, and terror, based on the instrumental nature of violence, which allows her to think about the originality of totalitarian domination. The objective of this intervention is to understand Hannah's opposition theorized between violence and terror, in order to see how this opposition proves fruitful to think about the colonial phenomenon and the development of racism. This intervention will focus on the functioning of slavery societies to show how they are already based on the "iron circle" described by Arendt to designate the totalitarian terror. The challenge then is to see how the arendtian conception of violence allows us to understand the role played by race in the construction of the modern state and its persistence in contemporary society.

Text

I wanted to speak about the arendtian distinction between power, violence and terror.

And see how these distinctions can be used to understand modern political facts.

Hannah Arendt is indeed a thinker who has tried to get back politic its own dignity. By opposition with thinkers like Max Weber who used to define power as a State monopole of legitimate physical violence, Hannah Arendt tried to understand the nature of power without taking violence into account. However, as she Understands power separately of violence, as two different concepts, she subsequently must redefine violence as an autonomous concept.

Firstly, Power is understood as the ability of being active with other men. Hannah Arendt describes power as a potentiality present each time people are gathering, speaking and acting with one another. Every power is legitimate by the only way to be an active community of men with their own rules, constitutions and dynamics.

Power is the process by which men are making things together. No matter the nature of the gathering. This can be a legal institution, a secret society, an illegal syndicate fighting for civil rights or even a knitting association working together to have access to municipal local.

In all these cases, there is a community able to engage and influence public space. Moreover, the community becomes its own public space where people can speak and act with other ones.

Power is the foundation of politic.

On the other hand, Hannah Arendt describes violence as a force used upon other men in order to insure a kind of domination. When one uses violence, he is never with other people but only against them.

For Hannah Arendt, violence is unpolitical, even better, it is anti-political. To use violence means to break up with the plurality of men and to establish a situation in which only strength prevails.

Violence doesn't care about other people's desire or willing. It is not a balanced relationship. Moreover, violence is not affected by number or masses. Like Arendt says, few people with machine guns can easily overcome thousands of pacific people. That's why a small military elite frequently dominates a whole disarmed population.

By opposition with power, violence is furthermore determined by its finality. When power is a process that is its own finality, violence must have a purpose. In the arendtian conception, violence never can be legitimate but can be justified by the goal to reach.

Violence is a tool and can be necessary used in certain circumstances. To paraphrase Arendt, violence is as useful for its holder as crime serves a robber. Thus, a dictator can use violence to stop a demonstration, a slave can use violence to escape his master etc. etc. That's why violence is so often used in political situation despite its unpolitical nature and that's why power, in its arendtian conception, is often confused with violence. Violence stops discussion, negotiation and compromise to instore a pure balance of power in which there is no need to deal with the other.

The problem, as Arendt says, is that if violence can easily destroy political power is unable to establish a new one. To build a new political order necessary leads to stop violence and get back to the negotiation table. In a sense, Arendt is quite close of the Clausewitzian comprehension of war. Not because Arendt would say that violence should be the continuation of policy with other means, but because she insists on the necessary limited character of violence.

Violence has to be framed by the political power that can only give it its purposes. By doing so, political space is temporarily abolished in favor of an opposition between opponents. However, the goal achievement also implies that politic takes back its rights.

Conversely, the impossibility to find back a political solution means to stay locked into an endless conflict. Ludendorff's attempt to reverse Clausewitz's formula by subjecting politics to the logic of war is a misunderstanding of both nature of politic and violence. As illustrated by German successive failures in the two world wars.

So, there are two independent concepts, nevertheless connected together. From now, the concept of terror is described as a confusion of the two precedent notions. Terror is a system in which violence is deliberately used in order to perpetuate the violence of the system. Like power, terror is distinguished by its lack of finality. The purpose of terror is only the perpetuation of itself, as a permanent state of violence. No longer limited by political issues, it extends to the whole public and social life. Having become its own principal, terror cannot simply be stopped by the victory of one side or the other but appears as a perpetual quest of real or imaginary opponents.

To illustrate the opposition between violence and terror, Hannah Arendt distinguishes the "desert of tyranny" from the "iron circle of terror". That means the violence destroys public spaces, leaving only a political vacuum characterized by powerlessness of citizen. On the other hand, terror encapsulates all the aspects of political and life in a state of permanent agitation in order to hide the lack of power.

Terror is a kind of radicalization where there is no more separation between political power and the use of violence. Terror is a dynamics of destruction the whole political and social life in a permanent movement. For Hannah Arendt, terror as a system is a characteristic of totalitarism. However, we can find other political situation, in which violence replace the dynamic of power and specially during the colonization that is presented by Arendt as an origin of totalitarism.

Hannah Arendt is indeed one of the first thinkers who analyses the link between imperialism and totalitarism. In this perspective, construction of Race as a ideologic system of domination has a major role in the construction of the totalitarian structure.

For Arendt, the construction of Race as a system of thinking and domination take place into the confrontation between white colonizer and Africa during the last part of the 19e century, and specially in south Africa. Inspiring by "Hearth of Darkness" from Conrad, she understands Race as a system of domination that become its own finality. She described Afrikaner's situation as an alienation-from-the-world in which there are no longer able to constitute a human world made of men action. They are not able to organize a new political order based on equality with black and the only way to maintain their racial domination is from now to continue to use violence without being able to stop it. Thus, colonization and race domination, are presented as the premises of a terror system There is many to say about this interpretation, and specially about the racism of Arendt's description of Africa as savage continent. Like Katheryn T. Gines said, it looks like Arendt "presents black people as the problem rather than situating white people anti-Black racism as the problem". Moreover, Arendt underestimates the role of race during the construction of European modernity.

Following here interpretation, there is a difference between the first colonization of the 18 th century where race was only a kind of prejudice without being a system of interpretation and the imperialism characterized by an endless process of colonial conquest. However, this analysis is only working because she is focused on the metropolitan situation without take colonies into account.

Of course, Arendt talked about slavery, but she doesn't analyze structure of modern slavery society, where slave could represent around 80% of the population. The plantation system already has all the characteristics of colonial society described by Arendt as a source of totalitarianism:

The system is indeed characterized by a permanent violence, illustrated by a very high mortality with around 60% people dying during the first year of their arrived. It's not a instrumental and limited violence used to reach a goal, but a paranoiac system that is building his own opponent in order to maintain his racial domination. Colonizer are always living into the fear of a black rebellion that could instore a black kingdom and reverse the racial domination. Fear that is a pure fantasy because black people only wanted to live in peace on their own land like we can see at the end of the Suriname's rebellion.

As we can see, there is no break between a first colonization period and a real imperialist moment. The plantation is already a system of terror based on a racial domination that reject equality of power. When Race become an ideological structure it establishes a pure state of terror that is beyond the simple use of violence.


Sulaiman Abo Diab
email: sabodiab@tishreen.edu.sy

Worked Examples on Using the Riemann Integral and the Fundamental of Calculus for Integration over a Polygonal Element

Keywords: Polygonal element, Fundamental of Calculus, Riemann integral, moments of area, stiffness matrix

In this paper, the Riemann integral and the fundamental of calculus will be used to perform double integrals on a polygonal domain enclosed by curved edges. The double integral with two variables over the domain is transformed into sequences of single integrals with one variable of its primitive. The sequence is arranged counter clockwise starting from the minimum value of the variable of integration. Finally, the integration over the domain is performed using only one variable in one direction. The way of integration is illustrated on practical examples for which the area and moments of area are found for arbitrary polygons enclosed by straight edges as well as curved edges and compared with the exact values resulting in from dividing the polygon into its standard elementary shapes and the parallel axis theorem. The stiffness matrix is derived for an arbitrary quadrilateral finite element for plate bending. The derived element is a generalization of the first finite element used in the analysis of thin plates known as ACM. The results are tested according to a program code written in MATLAB. The presented way of integration is of general applicability for convex and non-convex domains for a wide range of engineering and physical problems provided that, the primitive of the integrated function exists and it is continuous on the partial intervals of integration. The generalization of this technique to volume integrals over polyhedral domains is possible.

Introduction

The finite element methods among other numerical methods experienced significant developments after the use of the Voronoi Diagram in partitioning of a plane points into convex polygons [START_REF] Weisstein | Voronoi Diagram[END_REF]. Based on Voronoi Diagram, there were several mesh generator for polygonal and polyhedral elements with topology optimization. These offer a general framework for finite element discretization and analysis, see for example [START_REF] Talischi | Polymesher: a general-purpose mesh generator for polygonal elements written in Matlab[END_REF] and the mesh generators mentioned therein.

In the past fifteen years, many works have been published which operate on polygonal elements in the framework of numerical analysis and used intensively in the fields of applied engineering and physical sciences and even in medical and biological sciences [3, 4, 5, 6, 7, 8, 9, 10,]. An overview of previous developments on conforming polygonal and polyhedral finite elements is included in [START_REF] Beirão Da Veiga | High-order Virtual Element Method on polyhedral meshes[END_REF], and an overview on the use of different generalized barycentric coordinates in Galerkin finite element computations is included in [START_REF] Manzini | New perspectives on polygonal and polyhedral finite element methods[END_REF]. Several other papers that use the polygonal and polyhedral elements in different fields of computational Engineering are listed in [START_REF] Chi | Some basic formulations of the virtual element method (VEM) for finite deformations[END_REF]. Therefore, it was necessary to use flexible techniques in performing the integrals of the state variables defined on the domains analyzed. These techniques become necessary when changes occur suddenly in the geometry of the domains such as the appearance of cracks or ruptures within them. Some recent publications [11, 12, 13 14, 15] show that the topic is still under study and development; some others [START_REF] Shen | Biomechanics of a moth scale at ultrasonic frequencies[END_REF][START_REF] Kappert | An interactive surgical simulation tool to assess the consequences of a partial glossectomy on a biomechanical model of the tongue[END_REF][START_REF] Ferguson | Polygonal Virtual Element Spatial Discretisation Methods for the Neutron Diffusion Equation With Applications in Nuclear Reactor Physics[END_REF] reflect its use and wide spreading under different disciplines.

Concurrently, new software languages have been developed that are highly capable of meeting the requirements of researchers in conducting symbolic and arithmetic operations in a built-in software environment, including but not limited to MATLAB, Python, Julia and Octave, etc... [19,20,21,22,23,[START_REF] Meurer | SymPy: symbolic computing in Python[END_REF] By adopting the Riemann integral, the Green's formula or the Gauss divergence theorem in the calculation of integrals over complex domains by the use of open programming languages that use symbolic operations, young researchers are provided with powerful tools that can be used to address various types of physical and engineering problems.

Integration over a finite element of various shapes is an important part of every finite element code. The numerical integration consumes considerable part of the computational time. Therefore, developing explicit element matrices will reduce the computational costs considerably. One of the great advantages of using the Riemann integral is the possibility to develop explicit expressions of the element matrices at first symbolically and incorporating it after that in a numerical program code.

A study about the history of integration can be found in [START_REF] Hammarström | Origins of integration[END_REF], [START_REF] Bastian | An introduction to the generalized Riemann integral and its role in undergraduate mathematics education[END_REF]. Many classical examples exploiting the basic concept of the fundamental of calculus are presented in [START_REF] Edelstein-Keshet | Integral Calculus with Applications to the Life Sciences[END_REF]. According to [START_REF] Antonietti | Fast Numerical integration on polytopic meshes with applications to discontinuous Galerkin finite element methods[END_REF], the most employed technique of integration over polygonal and polyhedral elements is performed by subdividing them into standard-shaped elements and after that applying the corresponding integration rules on each sub-element and summation. [START_REF] Antonietti | Fast Numerical integration on polytopic meshes with applications to discontinuous Galerkin finite element methods[END_REF], itself presents quadrature rules for the numerical approximation of integrals of polynomial functions over general polygonal/polyhedral elements without subdivision exploiting the 'Stokes' Theorem. The surface or volume integral over a polygon is evaluated by computing the integral of the same function over the boundary. Although the integration does not require an explicit construction of a sub-tessellation into standard-shapes, the numerical integration used is complex and a mapping procedure is employed in deriving the stiffness and mass matrix. A limited exact integration technique for exact geometrical representation of the holes within the context of XFEM utilizing Fubini's Theorem can be found in [START_REF] Perumal | Analysis of thin plates with holes by exact geometrical representation within XFEM[END_REF].

This paper demonstrates some worked examples on integration over arbitrary polygonal domains enclosed by a sequence of edges. The integration is performed in the Cartesian coordinate system using only one variable. There is also no necessity to use any interpolation or to use any natural coordinate system. A mapping procedure is also not necessary. Once, the primitives of the functions to be integrated over the domains are known, the integration can be performed along the boundary between the limits in one direction using one variable. Furthermore, there is no need to use the Gauss divergence theorem, the 'Stokes' Theorem or to integrate in the edge direction. The method of integration is easy to use. It is presented in a form such that it can be adopted directly within a computer program for numerical analysis. Areas and moments of areas of a triangle, an arbitrary quadrilateral and a polygon enclosed by six straight edges as well as an arbitrary curved domain are calculated using the scheme presented and compared with the results, provided using the combination of these known values for standard shapes and using the parallel axes theorem. In addition, the integration of the stiffness matrix of a generalized version of the well-known ACM plate-bending element of quadrilateral shape is presented. The presented procedure is of general applicability for elements with curved edges and not limited to straight-sided edges in the framework of numerical methods. This work is mainly devoted to students and young researchers and therefore detailed calculations and program codes are listed.

Integration in the Cartesian coordinate system

Let  be a polygonal domain related to a Cartesian coordinate system . Let  be enclosed in the rectangle ( ) ;

) ( ) ( ) 2 ( ) 1 ( b a y y y x x x    
and be bounded by

n-edges ) 1 ( E , ) 2 ( E ,…, ) (n E described counterclockwise through the explicit sequence of equations ) 1 ( E : ] , [ ); ( ) 2 ( ) 1 ( 1 x x x x g y   (1.a) ) 2 ( E : ] , [ ); ( ) 3 ( ) 2 ( 2 x x x x g y   (1.b) ….. ) (n E : ] , [ ); ( ) 1 ( ) ( x x x x g y n n   (1.n)
The edges are connected by a sequence of vertices (1), (2) (…), (n) (nodal points) with the nodal coordinates

) ( p x , see Figure 1.                ) ( ) ( ) 2 ( ) 2 ( ) 1 ( ) 1 ( ) ( . .
     (2)
The Cartesian variables of ) , ( y x p are connected through the Pythagorean Theorem: This fact should not be ignored, when we change from a physical domain to a computational domain. For example, the Pythagorean Theorem still holds when changing x and y of the physical domain through the polar variables r and  as a computational domain.

2 2 2 2 2 y x r y x r      (3)
x y

) 1 ( ) 2 ( ) 4 ( ) 3 ( ) 2 (  n A ) 1 ( A ) 2 ( A ) (i E ) (i A ) (n E ) 1 (  n A ) 1 ( E ) 2 ( E ) 3 ( E ) 1 (  n E ) 2 (  i x ) 4 ( x ) 2 ( x ) 3 ( x ) 1 ( x ) (n x ) 1 (  i ) 2 (  i ) (n ) 1 (  i x ) (a y ) (b y
In Eqn. (3), there are only two independent variables. The variables x and y are variant in what concerns axes translation and rotation. The third one r , the distance between the point p and the origin o, is invariant and independent of the coordinate system used. Solving Eqn. ( 3 x and y , then these can be performed using only one variable as states in every encyclopaedia for mathematics, see for instance [START_REF]EncyclopediaofMathematics[END_REF]31]. Using for example

x as a variable, the following integral over the total area of Fig. 1:

  A dxdy y x f I ) , ( (7)
can be calculated as follows:

Suppose that ) , ( y x f is continuous and integrable over the subintervals [ )

1 ( x , ) 2 ( x ],…, [ ) 1 (  n x , ) (n x ],[ ) (n x , ) 1 ( x ] .
The integral 1 I over the subarea )

(

A enclosed between )

1 (

E and )

2 ( E , and the coordinate lines

) 3 ( x x  ) 2 ( x x  reads dx y x F dx y x F dx y x F y x F dx dy y x f I x x E x x E x x E E x x x g x g            ) 2 ( ) 3 ( 1 ) 3 ( ) 2 ( 2 ) 2 ( ) 3 ( 1 2 ) 2 ( ) 3 ( 2 1 )| , ( )| , ( ) )| , ( )| , ( ( ) ) , ( ( ) ( ) ( 1 (8)
The integral 2 I over the subarea ) 2 (

A

enclosed between )

1 (

E and )

3 ( E , and the coordinate lines

) 4 ( x x  ) 3 ( x x  reads dx y x F dx y x F dx y x F y x F dx dy y x f I x x E x x E x x E E x x x g x g            ) 3 ( ) 4 ( 1 ) 4 ( ) 3 ( 3 ) 3 ( ) 4 ( 1 3 ) 3 ( ) 4 ( 3 1 )| , ( )| , ( ) )| , ( )| , ( ( ) ) , ( ( ) ( ) ( 2 (9)
The integral i I over the subarea

) (i A enclosed between ) 1 (

E and )

(i E , and the coordinate lines

) 2 (   i x x ) 1 (   i x x reads dx y x F dx y x F dx y x F y x F dx dy y x f I i i i i i i i i i i i x x E x x E x x E E x x x g x g i                     ) 1 ( ) 2 ( 1 ) 2 ( ) 1 ( ) 1 ( ) 2 ( 1 ) 1 ( ) 2 ( 2 1 )| , ( )| , ( ) )| , ( )| , ( ( ) ) , ( ( ) (
) ( [START_REF] Wriggers | A virtual element method for contact[END_REF] And so on. For the Last part of the area ) (n

A enclosed between ) 1 (

E and )

(n E , and the coordinate lines

) (n x x  ) 1 ( x x  dx y x F dx y x F dx y x F y x F dx dy y x f I n n n n n n n x x E x x E x x E E x x x g x g n             ) ( ) 1 ( 1 ) 1 ( ) ( ) ( ) 1 ( 1 ) ( ) 1 ( 1 )
| , ( )| , ( ) )| , ( )| , ( ( ) ) , ( ( ) ( ) ( ) 1 ( (11)
Eqn. [START_REF] Chi | Some basic formulations of the virtual element method (VEM) for finite deformations[END_REF]

E , do not intersect in the subinterval   ) 3 ( ) 2 ( , x x
otherwise the subinterval or even the domain requires further subdivision in order to consider the intersection points between them. Similar conditions apply for Eqn. ( 9) to [START_REF] Chi | A paradigm for higher-order polygonal elements in finite elasticity using a gradient correction scheme[END_REF]. These conditions do not restrict the use of the method but they make its use more difficult.

In Eqn. [START_REF] Chi | Some basic formulations of the virtual element method (VEM) for finite deformations[END_REF] with respect to y . ) (x  is an arbitrary function independent of y . Selecting the integration of ) (x  along the closed interval as zero function and observing that

dx y x F dx y x F dx y x F dx y x F dx y x F n i i x x E x x E x x E x x E x x E              ) ( ) 1 ( 1 ) 1 ( ) 2 ( 1 ) 4 ( ) 3 ( 1 ) 3 ( ) 2 ( 1 ) 2 ( ) 1 ( 1
)| , ( )| , ( .... )| , ( )| , ( )| , ( (13)
Then, the definite integral can be performed with one variable with respect to x as a series of integrals over the subintervals counter clockwise using the following relation

       n i E x x A dx y x F dxdy y x f i i i 1 | ) , ( ) , ( ) 1 ( ) ( (14

(p)

i ~

A similar relation applies for the integration in the y -direction but now with a positive sign.

  

  n i E E A dy y x F dxdy y x f i i 1 | ) , ( ) , ( (15)
Now, let us explain how to employ this result. For the sake of simplicity, consider the arbitrary polygon depicted in Fig. 2. In case of a domain with straight edges described by their nodal points, the relation between For a quadrilateral domain for example, the domain is bounded by four edges E : 0

1 1 1    c y b x a (16.a) ) 2 ( E : 0 2 2 2    c y b x a (16.b) ) 3 ( E : 0 3 3 3    c y b x a (16.c) ) 4 ( E : 0 4 4 4    c y b x a (16.d)
In such case, one of the variables can be explicitly expressed in terms of the other one and the integration can be performed in one direction. In the following, integrals for calculating geometrical properties like area and moments of area of a triangle, of an arbitrary quadrilateral and of a polygon enclosed by six or five straight edges as well as of domains enclosed by curved edges will be demonstrated in details.

Integration over a triangular element using one variable

Example 1: A triangular domain is defined by its four vertices (1), (2) (3) (nodal points)

           4 . 2 8 . 1 0 . 0 0 . 5 0 . 0 0 . 0 ~) ( i p x
The exact values of the area of the triangular element and the moments of area about the axes ) , ( y x , computed by the usual way of integration about the axes of a standard shape (here, triangle) as well as the parallel axes theorem [START_REF] Timoshenko | Engineering Mechanics[END_REF] Now, the same results can be obtained using the above stated method of integration. The triangular element is bounded by three edges

        A y dA x I 32 . 10 (2.4) 3 1 (3.2)) 3 1 8 . 1 )( 2 ) 2 . 3 )( 4 . 2 ( ( 72 ) 2 . 3 ( ) 4 . 2 ( (2.4) 3 1 (1.8) 3 2 ( 2 ) 8 . 1 )( 4 . 2 ( 72 ) 8 . 1 ( ) 4 . 2 ( 2 2 2 2         A xy xydA I
E : 0  y 2 E : 4 15 4 3    x y 3 E : x y 3 4  Dividing the triangle in two subareas ) 2 ( ) 1 ( , A A
and observing the limits of the integral, the area and the moments of area about x -axes take the following form:

The area A of the triangle calculated as a sum of two subareas, the subarea

E dx dy dx dy dA A E E E E A         5 8 . 1 8 . 1 0 2 1 3 1 ) ( ) ( x y ) 1 ( ) 2 ( ) 3 ( ) 2 ( A ) 1 ( A A ) 1 ( E ) 2 ( E ) 3 ( E 6 | ) ( | ) ( | ) ( | ) ( 5 8 . 1 2 5 8 . 1 1 8 . 1 0 3 8 . 1 0 1           dx y dx y dx y dx y A E E E E 6 ) 3 4 ( ) 0 ( ) 4 15 4 3 ( ) 0 ( 8 . 1 0 8 . 1 0 5 8 . 1 5 8 . 1             dx x dx dx x dx A
The previous integral arranged counter clockwise is then as follows

          dx x dx x dx A
The moments of area x I calculated as the moment of area of the subarea )

(

A enclosed between the two edges )

1 (

E and

E E E E A x         5 8 . 1 2 8 . 1 0 2 2 2 1 3 1 ) ( ) ( 5.76 3 | ) ( 3 | ) ( 3 | ) ( 3 | ) ( 5 8 . 1 2 3 5 8 . 1 1 3 8 . 1 0 3 3 8 . 1 0 1 3           dx y dx y dx y dx y I E E E E x
The previous integral arranged counter clockwise is then as follows

                    dx x x dx x dx x x dx x dx y x dx y x dx y x dx y x I E E E E y
The previous integral arranged counter clockwise is then as follows

                     dx x x dx x dx x x dx x dx y x dx y x dx y x dx y x I E E E E xy
This gives arranged counterclockwise the same result. The calculation using the values about the axes of the standard shapes and the parallel axes theorem gives: Integration over a polygonal element using one variable Example 3: A polygonal domain is defined by its six vertices (1), (2) (3), ( 4), ( 5), (6) (nodal points)

x y ) 1 ( ) 2 ( ) 4 ( ) 3 ( ) 2 ( A ) 3 ( A ) 4 ( A ) 5 ( A ) 6 ( A ) (q A ) 1 ( A ) 1 ( E ) 2 ( E ) 3 ( E ) 4 ( E . 21 ) 2 ) 1 )( 4 ( 2 ) 1 )( 4 ( ) 1 )( 2 ( 2 ) 2 )( 4 ( 2 ) 1 )( 7 ( ( ) 5 )( 7 ( ) ( ) 6 ( ) 5 ( ) 4 ( ) 3 ( ) 2 ( ) 1 (              A A A A A A A 3 146.083333 (4)) 3 2 1 )( 2 ) 4 )( 1 ( ( 36 ) 4 )( 1 ( ) 3 2 4 )( 2 ) 1 )( 4 ( ( 36
) 1 )( 4 ( ) 2 1 4 )( 1 )( 2 ( 12 ) 1 )( 2 ( - (4)) 3 2 )( 2 ) 4 )( 2 ( ( 36
) 4 )( 2 ( ) ) 3 1 )( 1 )( 7 ( 36 ) 1 )( 7 ( {( - ) 2 5 )( 5 )( 7 ( 12 ) 5 )( 7 ( } { - 2 3 2 3 2 3 2 3 2 3 2 3 2 2 2 2 2 2 2 2 ) 6 ( ) 5 ( ) 4 ( ) 3 ( ) 2 ( ) 1 (                               A A A A A A A A x dA y dA y dA y dA y dA y dA y dxdy y dA y I 3 439.583333 ) 3 1 1 ) ( 2 ) 1 )( 4 ( ( 36 ) 1 )( 4 ( (4)) 3 2 2 )( 2 ) 4 )( 1 ( ( 36 ) 4 )( 1 ( ) 1 6 )( 2 )( 1 ( 12
) 2 )( 1 ( - (2)) 3 2 6 ) ( 2 ) 2 )( 4 ( ( 36 ) 2 )( 4 ( ) ) 3 7 1 (( 2 ) 7 )( 1 ( 36 ) 7 )( 1 ( {( - ) 2 7 1 )( 7 )( 5 ( 12 ) 7 )( 5 (
} { - 2 3 2 3 2 3 2 3 2 3 2 3 2 2 2 2 2 2 2 2 ) 6 ( ) 5 ( ) 4 ( ) 3 ( ) 2 ( ) 1 (                                 A A A A A A A A y dA x dA x dA x dA x dA x dA x dxdy x dA x I 196.875000 (4)) 3 2 )(1 3 1 1 )( 2 ) 1 )( 4 ( ( 72 ) 1 )( 4 ( ) 3 2 4 ( (4)) 3 2 2 )( 2 ) 4 )( 1 ( ( 72 ) 4 )( 1 ( ) 2 1 4 ( ) 1 6 )( 2 )( 1 ( 0 - ) 4 ( 3 2 ( (2)) 3 2 6 )( 2 ) 2 )( 4 ( ( 72 ) 2 )( 4 ( ) ) 3 1 ( ) 3 7 1 (( 2 ) 7 )( 1 ( 72 ) 7 ( ) 1 ( {(- - ) 2 7 1 )( 2 5 )( 7 )( 5 ( 0 } { - 2 2 2 2 3 2 2 2 ) 6 ( ) 5 ( ) 4 ( ) 3 ( ) 2 ( ) 1 (                                    A A A A A A A A
) 8 7 / ( 3 | ) ( 3 | ) ( 3 | ) ( 3 | ) ( 3 | ) ( 3 | ) (
                                      dx x
                 dx x dx x dx x dx x I x 33 439.583333 ) 16 2 ( ) 8 7 / ( ) 4 / 22 4 / ( ) 8 7 / ( ) 3 4 ( ) 8 7 / ( | ) ( | ) ( | ) ( | ) ( | ) ( | ) (
                                     
                 dx x x dx x x dx x x dx x x I y 196.875 2 ) 16 2 ( 2 ) 8 7 / ( 2 ) 4 / 22 4 / ( 2 ) 8 7 / ( 2 ) 3 4 ( 2 ) 8 7 / ( 2 | ) ( 2 | ) ( 2 | ) ( 2 | ) ( 2 | ) ( 2 | ) (
                                     

A

with the side length 6 8 and the summation of them for four rectangled triangles

) 5 ( ) 4 ( ) 3 ( ) 2 ( , , , A A A A with the catheti 2 1 , 3 4  , 3 2  and 4 3 , Fig. 4 ( [32]) 32 ) 2 ) 4 )( 3 ( 2 ) 3 )( 2 ( 2 ) 3 )( 4 ( 2 ) 2 )( 1 ( ( ) 6 )( 8 ( ) ( ) 5 ( ) 4 ( ) 3 ( ) 2 ( ) 1 (            A A A A A A 353.833333 (4)) 3 2 2 )( 2 ) 4 )( 3 ( ( 36
) 4 )( 3 ( (3)) 3 2 3 ( 2 ) 3 )( 2 ( 36 ) 3 )( 2 ( - (3)) 3 1 ( 2 ) 3 )( 4 ( 36
) 3 )( 4 ( ) ) 3 2 ( 2 ) 2 )( 1 ( 36 ) 2 )( 1 ( {( - ) 2 6 )( 6 )( 8 ( 12 ) 6 )( 8 ( } { - 2 3 2 3 2 3 2 3 2 3 2 2 2 2 2 2 2 ) 5 ( ) 4 ( ) 3 ( ) 2 ( ) 1 (                          A A A A A A A x dA y dA y dA y dA y dA y dxdy y dA y I x y ) 2 ( ) 3 ( ) 6 ( ) 4 ( ) 2 ( A
) 3 ( A ) 4 ( A ) 4 ( E ) 5 ( A ) 6 ( E ) 1 ( A ) 1 ( E ) 2 ( E ) 3 ( E ) 5 ( E ) 1 ( ) 5 ( 580.833333 (3)) 3 1 )( 2 ) 3 )( 4 ( ( 36
) 3 )( 4 ( ) 3 4 6 ( 2 ) 3 )( 2 ( 36 ) 3 )( 2 ( ) 3 8 4 )( 2 ) 3 )( 4 ( ( 36
) 3 )( 4 ( ) 3 1 ( 2 ) 2 )( 1 ( 36 ) 1 )( 2 ( {( - ) 2 8 )( 8 )( 6 ( 12 ) 8 )( 6 ( } { - 2 3 2 3 2 3 2 3 2 3 2 2 2 2 2 2 2 ) 5 ( ) 4 ( ) 3 ( ) 2 ( ) 1 (                          A A A A A A A y dA x dA x dA x dA x dA x dxdy x dA x I 394.333333 = Ix (4)) 3 2 (3))(2 3 1 )( 2 ) 4 )( 3 ( ( 72 ) 4 )( 3 ( )) 3 ( 3 2 3 ( (2)) 3 2 6 )( 2 ) 3 )( 2 ( ( 72
) 3 )( 2 ( )) 3 ( 3 1 ( (4)) 3 2 4 )( 2 ) 2 )( 4 ( ( 72 ) 3 )( 4 ( ) ) 3 2 ( ) 3 1 (( 2 ) 2 )( 1 ( 72 ) 2 ( ) 1 ( {(- - ) 2 6 )( 2 8 )( 6 )( 8 ( 0 } { - 2 2 2 2 2 2 2 2 ) 5 ( ) 4 ( ) 3 ( ) 2 ( ) 1 (                            A A A A A A A xy xydA xydA xydA xydA xydA xydxdy xydA I
The polygonal domain is bounded by six edges

6 5 4 3 2 1 , , , , , E E E E E E
. Every edge-equation can be determined by the two corresponding vertices spanning the edge. The resulting edge-equations solved with respect to y are as follows

E : 2 2    x y 2 E : 0  y 3 E : 3 4 3   x y x y ) 1 ( ) 3 ( ) 6 ( ) 4 ( ) 2 ( A ) 3 ( A ) 4 ( A ) 5 ( E ) 1 ( A ) 1 ( E ) 2 ( E ) 3 ( E ) 4 ( E ) 6 ( E ) 5 ( ) 2 ( x ) 6 ( x ) 3 ( x ) 5 ( x ) 1 ( x ) 4 ( x ) 5 ( A ) 2 (
                                                                           
E E E E E E E E E E E E E E E E E E E E E E E E E E A A A A A A
In an analogous way, the moments of area about

x -axes take the form This can be written in the following form: 394.333333 2

E E E E E E E E E E E E E E E E E E E E A A A A A A x                                               
| ) ( 3 | ) ( 3 | ) ( 3 | ) ( 3 | ) ( 3 | ) ( 3 | ) ( 3 | ) ( 3 | ) ( ) ( ) ( ) ( ) ( ) ( 3
                                 dx x
E E E E E E E E E E E E E E E E E E E E A A A A A A y                                               
2 2 2 2 2 2 2 | ) ( | ) ( | ) ( | ) ( | ) ( | ) ( | ) ( | ) ( | ) ( | ) ( ) ( ) ( ) ( ) ( ) (
                                 dx x
E E E E E E E E E E E E E E E E E E E E A A A A A A xy                                               
2 | ) ( 2 | ) ( 2 | ) ( 2 | ) ( 2 | ) ( 2 | ) ( 2 | ) ( 2 | ) ( 2 | ) ( 2 | ) ( ) ( ) ( ) ( ) ( ) (
) 2 3 / 4 ( 2 ) 6 ( 2 ) 15 2 / 3 ( 2 ) 3 4 / 3 ( 2 ) 0 ( 2 ) 2 2 ( 2 | ) ( 2 | ) ( 2 | ) ( 2 | ) ( 2 | ) ( 2 | ) (
                                 dx x x dx x dx x x dx x x dx x dx x x dx y x dx y x dx y x dx y x dx y x dx y x I E E E E E E xy
The same results can be obtained by dividing the polygon into two quadrilateral with the following nodal points and finding the explicit transformation relation between Cartesian variables and natural variables using the standard bilinear approach and integrating the expressions for area and moments of area between -1 and +1. In the following, a MATLAB function code is given by which the area and moments of area for an arbitrary polygon with straight edges can be calculated. Before using it, some notes on using formula ( 14) within a finite element program code must be considered. The vertices of the polygonal domain must be ordered counterclockwise starting with the minimum

x -value when integrating over

x or the smallest y -value when integrating over y in order to avoid any sign confusion which leads to incorrect results. Furthermore, an edge equation of the form const x  , when integrating over x or of the form const y  , when integrating over y must be avoided. In case of a parallel edge to x -axes the program will stop running from itself (the code integrate over

x ). The presented code is not intended for a general use in a finite element program because it needs further editing to account for some special geometry cases of a polygon. It should serve understanding and investigating the use of the Riemann formula in the current paper.

In order to call the code, the following script file must be saved under the name moa.m in the working directory. The same results can be obtained by dividing the polygon into two quadrilateral with the following nodal points

                          0 . 5 0 . 2 0 . 6 0 . 8 0 . 3 0 . 3 0 . 2 5 . 1 ; 0 . 3 5 . 1 0 . 3 0 . 3 0 . 0 0 . 6 0 . 1 0 . 1 ~) ( ~) ( i p i p x x
and performing the integration using the standard bilinear approach.

Here, the results are also exact in case of non-convex polygon.

Example 5:

The last assessment in this section is on a circular disc with a diameter of two units of length. The circumference is divided into hundred segments. The above function moa(xip) is used along with the following code: % area and moments of area for a circular disc with a diameter of two units of length

E ) 5 ( E ) 1 ( E ) 2 ( E ) 3 ( E ) 1 ( (p) i ~
These results are quasi correct (A=pi; Ix=Iy=pi/4 ; Ixy=0). It is worth mentioning that the convergence towards the exact results occurs very slowly especially for the quadratic form of Ix or Iy. One can easily discover this fact by changing the number of segments in the for loop.

Two reasons may delay the use of Riemann integral. Firstly, one needs to know the primitive of the function to be integrated inside the domain. Secondly one needs also to express one variable using the other one explicitly at the boundaries of the polygon, which is probably impossible in case of a complex geometry bounded by curves with complex implicit expressions in the two variables. In the second case, one can use one-dimensional curve fitting techniques to find explicit expressions at the domain boundaries.

7

Integration over domain with curved edges

   x y E 2 1 1 : x y E   1 4 : 2 4   x y E ) 1 ) 2 ( 4 : 2 3
A 5 A 4 A 3 A 1 A x y ] 1 , 0 [ ; ) 1 ( : 2 1    x x y E ] 2 , 1 [ ; 1 : 2    x x y E ] 2 , 1 [ ; 1 ) 2 ( 4 : 2 3     x x y E ] 1 , 0 [ ; 1 4 : 2 4    x x y E 1 ( ) 4 ( ) 3 ( ) 2 ( 2
In order to calculate the exact values of the area and the moments of area of the curved domain using standard elementary shapes, the five shapes shown in Fig. 7 are considered. Corresponding to the concept of the fundamental of calculus, the area of the curved domain can be calculated as follows

                 x x x x x x I I I I I I 3.9869838 )) 4 ( 4 1 ) 1 )( 1 ( 3 1 2 3 2 3 2 3 4 3 ) 5 ( ) 4 ( ) 3 ( ) 2 ( ) 1 ( 
                 y y y y y y I I I I I I 6.45 )) 1 ( 4 1 1 ))( 4 ( 10 3 1 )( 4 )( 1 ( 3 1 )) 1 )( 1 ( 4 3 ))( 4 )( 1 ( 10 3 1 )( 4 )( 1 ( 3 1 )) 1 ( 3 1 1 )(( 1 )( 3 2 ( 2 ) 1 )( 1 ( ) 1 ( ) 1 ( 1 1 )(( 4 )( 1 ( 3 1 ) 4 ( ) 1 ( 80 1 )) 1 ( 4 3 )( 4 )( 1 ( 3 1 ) 4 ( ) 1 ( 80 1 )) 1 )( 3 1 1 )(( 1 )( 1 ( 2 1 ) 1 )( 1 ( 36 1 ) 1 )( ( 16 1 ) 1 )( 1 ( 3 1 2 3 2 3 2 3 4 3 ) 5 ( ) 4 ( ) 3 ( ) 2 ( ) 1 ( 
1 ) 2 1 ( ) 1 ( 72
1 ) 2 1 )( 2 1 )( 1 )( 1 ( ) 2 2 2 2 ) 5 ( ) 4 ( ) 3 ( ) 2 ( ) 1 (                  8
(p) i ~ 3.3812685 ) 1 4 ( ) 1 ) 2 ( 4 ( ) 1 ( ) 1 ( | ) ( | ) ( | ) ( | ) ( | ) ( | ) ( | ) ( | ) ( ) ( ) (
                                          dx x dx x dx x
E E E E E E E E E E E E A A
Then, the moments of area read 15.5488885 The finite Element approximation is based on Hamilton's Principle. The 2D expression for the special case of the thin plate considered can be written in the absence of the prescribed boundary displacements relating to a Cartesian coordinate system in the following form:

                        dx x dx x dx x dx x dx y dx y dx y dx y I E E E E x 3.9869838 ) 1 ) 1 ( 4 ( ) 1 ) 2 ( 4 ( ) 1 ( ) ) 1 ( ( | ) ( | ) ( | ) ( | )
                         dx x x dx x x dx x x dx x x dx y x dx y x dx y x dx y x I E E E E y 6. 45 2 ) 1 4 ( 2 ) 1 ) 2 ( 4 ( 2 ) 1 ( 2 ) ) 1 ( ( 2 | ) ( 2 | ) ( 2 | ) ( 2 |
                        dx x x dx x x dx x x dx x x dx y x dx y x dx y x dx y x I E E E E xy
0 2 1 2 1 2 1 3 3 1 0 ) ( ) ( 0                 dt u F dA u u dA u q dA E t t n i i x i A A j ij i x kl A ijkl ij       (17)
where 1 t and 2 t are two fixed time points of the vibration process,  is the first variation, A is the area and dA its differential element, i u  is the velocity vector in which both displacement and rotation components are included,

j i
 is the corresponding mass density matrix, ) (i F is the concentrated load applied at the point (i). ij  is the curvature tensor, which reads expressed in terms of the deflection ) , (

E

is the matrix of the force-curvature dependency given in a matrix form as follows:

                  0 0 0 0 2 / ) 1 ( 2 / ) 1 ( 0 0 2 / ) 1 ( 2 / ) 1 ( 0 0 0 1 ) 1 ( 12 2 3        Eh E ijkl (19) j i
 is defined by the following matrix: In Eqn. [START_REF] Kappert | An interactive surgical simulation tool to assess the consequences of a partial glossectomy on a biomechanical model of the tongue[END_REF], the internal work associated with the bending and twist moments is only considered. Deriving the curvature tensor ij  using Eqn. [START_REF] Meurer | SymPy: symbolic computing in Python[END_REF] yield

                 2 1 2 1 2 2 2 1 2 1 2 2 2 1 2 1 2 1 2 1 ) ( ) ( 1 ) ( ) ( ) ( 6 0 2 0 0 2 0 0 0 0 0 ) ( 3 ) ( 3 0 2 2 0 0 1 0 0 0 0 ) ( 3 ) ( 3 0 2 2 0 0 1 0 0 0 0 0 6 0 0 2 6 0 0 2 0 0 0 ) ( x x x x x x x x x x x x x x x x p u A p m m r r m m r r m m ij ij ij  (25)
Applying the expressions of ij  in the first term of Eqn. [START_REF] Kappert | An interactive surgical simulation tool to assess the consequences of a partial glossectomy on a biomechanical model of the tongue[END_REF] gives [START_REF] Edelstein-Keshet | Integral Calculus with Applications to the Life Sciences[END_REF] is the element stiffness matrix related to the Cartesian coordinate system.

) ( ) ( ) ( ) ( ) ( 1 ) ( ) ( ) ( ) ( 1 ) ( ) ( ) ( 2 1 ) ) ) 2 1 2 1 ( ( ( q q q q r r r r q q n n q q n n ijkl A m m m m r r r r kl A ijkl ij u k u u A dA p E p A u dA E kl ij         (26) where 1 ) ( ) ( ) ( ) ( 1 ) ( ) ( ) ( ) ( ) ) ) ( ( (     n n q q n n ijkl A m m m m r r q q r r A dA p E p A k kl ij
The integration over the area can be performed using the scheme presented above as follow: This example shows the necessity to define the proposed local element coordinate system and to locate it at the geometric center of the element.

           n i E x x n n m m n n ijkl x m m x n n ijkl A m m dx P dx dx p E p dA p E p I i i i r kl ij kl ij 1 1 ) ( ) ( 1 2 ) ( ) ( ) ( ) ( | ) ( ) ) 1 ( ) ( 1 2 ( (28) where )
Similar procedure can be adopted to evaluate the element mass matrix resulting in from evaluating the third term of Eqn. [START_REF] Kappert | An interactive surgical simulation tool to assess the consequences of a partial glossectomy on a biomechanical model of the tongue[END_REF].

A final note about deriving the stiffness matrix of the rectangular ACM plate-bending element is to be mentioned. It is better to use a separate program-code for deriving this matrix. The Fundamental of Calculus can deal with this special case, too. For a rectangular element with the side-length b a l l  and the origin of the coordinate system located at the element center, the edges 2 E and 4 E lie on the coordinate lines a l x   ; a l x   , respectively. The following simple code integrated using the Fundamental of Calculus in its ideal form gives the exact stiffness matrix:

E(1)=-lb;E(3)=lb; kr=int(subs(k,y,E (3)
),-la,la)-int(subs(k,y,E(1)),-la,la)

where k is the primitive matrix of the matrix pT*c*p as constructed as in Ekr.m but now with the special nodal coordinates of the rectangular element.

As may be seen, formula [START_REF] Ho-Nguyen-Tan | Polygonal shell elements with assumed transverse shear and membrane strains[END_REF] or [START_REF] Wang | n-sided polygonal hybrid finite elements with unified fundamental solution kernels for topology optimization[END_REF] applies in case of rectangular element, too, but only two edges have to be considered during the integration.

Note that the results in the first seven examples using either formula [START_REF] Ho-Nguyen-Tan | Polygonal shell elements with assumed transverse shear and membrane strains[END_REF] or the standard bilinear approach are exact but the results of the stiffness matrix are different. In such cases, it is difficult to make right conclusions. A study on benchmark of polygon quality metrics for polytopal element methods is still at the beginning [START_REF] Attene | Benchmark of Polygon Quality Metrics for Polytopal Element Methods[END_REF]. However, one can observe that the characters of the integrals in the first five examples are additive, whilst they are not additive when deriving the stiffness matrix, which include the rotations and their derivatives. In addition the integrated shape functions are discontinuous and non-conform. Therefore, we found ourselves confronted with similar problems that appear during the discretization. Therefore, it is necessary to subject the element matrices to the same tests required for a finite element application especially when the integrated function are discontinuous.

8

Changing the variable of integration 8.1.

Using a polar coordinate system

An example of changing the variable of integration (i. e. x or y ) is the use of a polar coordinate system. In polar coordinate system the following exact relations apply In this sense, the transformation is exact and there is no approximation.

The position vector of an arbitrary point The deferential element of the area can now be calculated as a vector product of these changes

The integral over the area is now to be performed over the new variables Usually, integration is first performed over the variable limits and after that over the constant limits.

8.2.

Using a s ,

 system

In the previous examples both limits of the integral change in

x and y -direction even if the domain is bounded by straight edges. Sometimes it is useful to use another coordinate system with one variant variable instead of using the Cartesian coordinate system or a polar coordinate system. Therefore, it is more reliable to find other two variables instead of x and y such that at least one of them has a constant value at the edges of the domain. Along the boundary with straight edges both x and y change, but y has a constant value.

The idea now is to decompose the position vector r where,   is the position vector of an arbitrary point o of the plane 0  z and s  the vector from o to p .

o can be located everywhere in the plane 0  z .

An exact relation between the magnitude of s   ,  and of that of the position vector is then given by This relation is valid everywhere in the domain and along the boundaries with o as free selectable point.

In order to evaluate integrals such as  A dxdy or  A dxdy y x f ) , ( , y x, must be expressed in terms of s ,  .

In case of straight edges, after suitable selection of o , the relation between s ,  and y x, can be formulated at the boundary. The integral can be performed in the s ,  instead of y x, . The Wachspress space [START_REF] Wachspress | A Rational Finite Element Basis[END_REF] offered an approximation basis for many publications dealing with polygonal and polyhedral elements. This idea can be extended by decomposing the position vector r  into more than two vectors   and s  when necessary and useful. In any case the connection between the origin of the coordinate system y x, and o should not be lost.

Conclusion

This paper shows the power of the Fundamental of Calculus in dealing with complex geometry-domains for some practical engineering problems. The integration over a polygonal domain enclosed by a sequence of edges is performed exactly using the Fundamental of Calculus without sub-division. Double integrals are transformed into sequences of single integrals in one direction using the Fundamental of Calculus. There were no need to use the Gauss divergence theorem, the 'Stokes' Theorem or to integrate in the edge direction. It was not necessary to use any mapping procedure in order to integrate over a polygonal element. The presented way of integration is of general applicability for convex and non-convex domains for a wide range of engineering and physical problems. Some examples with curved edges are also investigated. The results are encouraging. Some notes on changing the variables of the integration for instance, in case of using polar coordinate system or non-orthogonal coordinate system are discussed.

  of the domain. The position vector of p reads

Figure 1 :

 1 Figure 1: Domain enclosed by curved edges, Cartesian coordinates of vertices

  determined depending on the nodal coordinates spanning the edge and the intersection point between the edge-line and the x -axes.

  edgeequation can be determined by the two corresponding vertices spanning the edge. The resulting edge-equations are as follows ) 1 (

Figure 2 :

 2 Figure 2: Triangular element, Cartesian coordinates of vertices, and edges

  edge-equation can be determined by the two corresponding vertices spanning the edge. The resulting edge-equations solved with respect to y are as follows 1

  as the moment of area of the subarea )

Example 2 :xFigure 3 :

 23 Figure 3: Quadrilateral cross-section, Cartesian coordinates of vertices, and edges

Figure 4 :

 4 Figure 4: Polygonal domain enclosed by six straight edges, Cartesian coordinates of vertices, and edges

Figure 5 :

 5 Figure 5: Polygonal domain enclosed by six straight edges, subintervals of integration

Example 4 :Figure 6 :

 46 Figure 6: Polygonal domain enclosed by five straight edges, Cartesian coordinates of vertices, and edges

  i=0; for f=-pi:pi/100:pi-pi/100 i=i+1 xip(i,1)=cos(f);xip(i,2)=sin(f); end xip [A,Ix,Iy,Ixy]=moa(xip) The output gives the following results A =3.1410759078128293940787214943951;Ix =0.78513981600761063777789906861322 Iy =0.78513981600761063432172486883439; Ixy =-0.0000000000000000116128775033486671270401753.

Figure 7 :Example 6 :

 76 Figure 7: Domain with curved edges, Cartesian coordinates of vertices, and edge-equations Example 6: Let a four-sided curved domain defined by its four vertices (1), (2) (3), (4) (nodal points).

Example 7 :

 7 Let a four-sided curved domain defined by its four vertices (1), (2) (3), by four curved parabolic segments . The edges are defined by the following edge-equations:

]

  The reader can easily verify these results by trying the following code %============================================================================ % Area and Moments of area for a domain bounded by four parabolic segments %============================================================================ syms x y %defining the quadrilateral curved domain xip=[1 1;8 0;6 4;2 5] %vertices %Edge equations Deriving the Stiffness matrix of a generalized ACM-plate bending element

  where  is the material density, E is the modulus of elasticity, h is the plate thickness and  the Poisson's ratioThe indicial notation to indicate the Cartesian variables 21 , x x is used instead of the y x, -frame and indices between brackets range over the nodal points.

x

  of the quadrilateral element of sec. 4 with the nodal coordinates . Defining a local coordinate system from the directions of the base vectors corresponding to[START_REF] Abo Diab | Generalization of a reduced Trefftz-type approach[END_REF],[START_REF] Abo Diab | Quadrilateral folded plate structure elements of reduced Trefftz type[END_REF], the new local coordinates readThe following two function code generate the stiffness matrix of the quadrilateral element %============================Ekr.m==============================% function [k]=EKr(xip,E,m,t) %xip: local cartesian coordinates of the quadrilateral %E: Young modulus m Poissons ratio t:thickness %===============================================================% syms x y ; M=[ 1 x y x^2 x*y y^2 x^3 x^2*y x*y^2 y^3 x^3*y x*y^3]; Mxx=diff(diff(M,x),x); Mxy=diff(diff(M,x),y); Myy=diff(diff(M,y),y); %c: eleasticity tensor in the x y-system c=((E*t^3)/(12.*(1-m^2)))*[1 0 0 m; 0 (1-m)/2. (1-m)/2. 0; 0 (1-m)/2. (1-m)/2. 0;m 0 0 1]; p=[Mxx;Mxy;Mxy;Myy]; Mi=[M;diff(M,y);-diff(M,x);]; A=[subs(subs(Mi,'x',xip(1,1)),'y',xip(1,2)); subs(subs(Mi,'x',xip(2,1)),'y',xip(2,2));subs(subs(Mi,'x',xip(3,1)),'y ',xip(3,2));subs(subs(Mi,'x',xip(4,1)),'y',xip(4,2))]; a=inv(A); k=transpose(p)*c*p %Primitive matrix of the matrix pT*c*p k=int(k,y) %integration over the boundary using x as variable [kr]=Rk(k,xip) k=transpose(a)*

y

  are replaced each by other two variables, namely  , r such that the Pythagorean Theorem applies.

  the position vector r  corresponding to the Chasles relation of vector addition as a sum of two vectors s

  )

	F	(	x ,	y	)	|	denotes the function	F	, ( y x	)	, in which	y is replaced by the explicit edge equation
							E
							i
	E	:	y		g	(	x	)	corresponding to Eqns. (1). This means, the edge equation is solved with respect to	y and the
	i					i
	expression	F	(	x ,	y	)	|	involves terms of x variables only.
													E
													i

Table 1 :

 1 Eigen values of the stiffness matrix of the quadrilateral element of sec. 4 related to a local Cartesian coordinate system (integrated exactly using the Riemann integral) Eqn. (27) between the limits -1, +1. The results of such procedure are listed cursive in the second line of Tab.1.Transforming the stiffness matrix into the global coordinate system gives the same results listed in Tab.1.

			kr*a
	eig(k)
	%==============================EOF====================================
	%==============================Rk.m===================================
	% function for calculating stiffness matrix of a Polygon with straight
	% edges using Riemann integral
	% before call order the x-coordinate such that x(1)=min
	%=====================================================================
	function [kr]=Rk(k,xip)
	n=size(xip);
	if (xip(1:1,1))< min(xip(2:n(1)-1,1))
	kr=zeros(12);
	syms x y
	for i=1:n(1)-1
	E(i)=det([ x y 1; xip(i,1) xip(i,2) 1; xip(i+1,1) xip(i+1,2) 1]);
	E(i)=solve(E(i),y);
	kE=subs(k,y,E(i));
	kr=kr-vpa(int(kE,xip(i,1),xip(i+1,1)));
	end
	E(i+1)=det([ x y 1; xip(n(1),1) xip(n(1),2) 1; xip(1,1) xip(1,2) 1]);
	E(i+1)=solve(E(n(1)),y);
	kE=subs(k,y,E(i));
	Kr=kr-vpa(int(kE,xip(n(1),1),xip(1,1)));
	vpa(kr,6);
	else
	fprintf('Please order the cordinate counter clockwise starting with
	the smallest xip-one');
	end
	end
	%===============================================================#
	Calling the above two functions using the following three commands
	xip =[ -3.2071 -1.5896; 3.8179 -2.3950; 1.7077 1.5480; -2.3185 2.4366]
	EKr(xip,1365.0,0.3,0.2)
	[kr]=Rk(k,xip)
	produces the stiffness matrix and its eigenvalues listed in the first line of Tab. 1.
	1 	2 	3 	4 	5 	6 	7 	8 	9 
	26877.01 7835.507 58177253 2807...3 68.57.07 68032631 68013573 68717373 68201772
	2286.126 78371301 58103713 28.67337 68.27230 68756320 68036..6 68751237 68203377

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Moving in x -axes positive direction, the area of the polygon can be integrated after dividing the area A of the polygon in five subareas )

A corresponding to the discontinuity points of the geometry as follows, see Fig. 5 32

A is the area of the unit quadrate with a unit side length. A is the area of a quarter bi-unit circle with a center located at the origin of the coordinate system . The area of the domain can now be calculated as follows

Then, the moments of area read 15.548889 ))

)) 4 ( 10 16The plate finite element with the nodal points (i), (j), (k), (l) has three degrees of freedom each node. These are the displacement normal to the plate surface in 3 x -direction and the two rotations about 1 x and 2 x -axes. The total number of degrees of freedom each element is then represented by the element nodal displacement vector with 12 degrees of freedom

The local axes 2 1 , x x are defined using the directions of the element base vectors in an analogous procedure used in section 4 in [START_REF] Abo Diab | Generalization of a reduced Trefftz-type approach[END_REF][START_REF] Abo Diab | Quadrilateral folded plate structure elements of reduced Trefftz type[END_REF]. The approximation basis is constructed using the defined local Cartesian variables